According to https://bluefringes.com/notable_people/r-moshe-shternbuch/
Rabbi Yisroel Barkin mentioned that one of R’ Shternbuch’s grandson’s wrote against Techeiles, and R’ Shternbuch signed off on it without doing the proper research. Now that he has done the research, he since then obtained a few Tallis Kottons and will wear them from time to time, but it’s not an everyday occurrence.
To quote R’ Avrohom Gross on tying and string ratio from an email: “He originally had Tosfos, but switched to Raavad because of the Gra in nistar, and he is a follower of the Gra (his shul is named after the Gra).”
From R’ Yechezkel Toporovitch:
בס”ד
אני עצמי סיפקתי להגאון ר’ משה שטרנבוך שליט”א פתילי תכלת עבור טלית קטן על ידי הבעל תוקע שלו הרב מיכאל לוי שליט”א וכן על ידי הבן של הראב”ד הרה”ג ר’ חיים עוזר שליט”א
למיטב ידיעתי הוא לובש מידי פעם שמעתי שיש תקופות שהוא לובש כל יום לפני התפילה
כמו כן שמעתי מבנו הגר”חע שאביו הראב”ד שליט”א אמר לו שאפשר לפרסם בשמו שהוא כבר לא נגד כמו שהיה פעם
לגבי שיטות מספר החוטים הנה בתחילה סבר לעשות כהתוס’ וכך גם שלחתי לו אח”כ אחרי שעיין בזה שוב החליף לשיטת הראב”ד
לגבי פרסום הדבר שיש אצלו ציצית עם תכלת ושהוא לובש מידי פעם, אפשר לשאול את בנו הנ”ל אם אפשר לפרסם
בכל הכבוד הראוי
יחזקאל משה הלוי טופורוביץ
פעיה”ק ירושלים תובב”א
Translated:
With siyata dishmaya,
I myself had the zechus to supply the gaon, Rav Moshe Sternbuch, shlit”a, with Techeiles strings for his tallis katan. I sent them through his baal toke’a, Rabbi Michael Levi, shlit”a, and also through the Ra’avad’s son, Harav Chaim Ozer, shlit”a.
As far as I know, he wears it from time to time, and I’ve heard that there are periods when he puts it on every day before davening. I also heard from the son, Rav Chaim Ozer, that his father, the Ra’avad, shlit”a, mentioned to him that it can now be publicized that he is no longer opposed to it as he once was.
Regarding the different shittos for the number of strings, originally, he leaned towards the Tosafos approach, and that’s how I sent it to him initially. Later on, after reviewing the sugya again, he switched to the Ra’avad’s approach.
As for publicizing that he has tzitzis with Techeiles and wears it occasionally, it would be best to check with his son mentioned above if it’s permissible to share this information.
With all due kavod,
Yechezkel Moshe HaLevi Toporowitz
Ir HaKodesh Yerushalayim, sheyibaneh v’shetechadesh
In a Hebrew article by Kikar Hashabat https://www.kikar.co.il/yeshiva-world/437107 it also listed Rav Shternbuch as wearing Tekhelet although it did not explain, how often he wears Tekhelet (Techeiles) and can it be seen visibly on his large Talit (Tallis).
See also Techeiles Response to Rabbi Kallus part 3 Questions and answers.mp4 where Rabbi Yisroel Rosenberg also claims within the video that Rav Moshe Shternbuch started wearing Tekhelet.
Comment: Excerpt from A Solution to Fundamental Problems Regarding Tekhelet
20 The Viewpoint of the Vilna Gaon On The Subject of the Number of Tekhelet Strings That One Should Put On
An Introduction
There are Rishonim, such as, Rambam, that ruled that one thread out of the eight (after it has been tied) will be Tekhelet and the rest will be white (or other colors).
There are Rishonim such as, Meiri, who held that two out of the eight threads be Tekhelet.
Rashi, Rabbeinu Tam, Rosh and Baal Ha’itur, and more, held that four out of the 8 threads be Tekhelet.
Rabbi Tzvi (Hershel) Schachter established that for practical halachic purposes the ruling is four out of eight threads.
As sources for his opinion he brought Mishna Brura, Chapter 9, siman katan 7 and the author of the Shulchan Aruch, to chapter 12, siman 1, on the matter of torn threads, who brought the opinion of Rabbeinu Tam*, as halacha, that we always use two threads of Tekhelet and two of white [before tying them into 8]. And so too, see the Mishna Brura, Chapter 9, siman katan 14, in the name of the Malbim, that with two threads [before tying] that are like the garment and two threads that are made from wool, he has done properly, for there are two separate parts to the commandment regarding Tzitzit, two strings of white and two strings of Tekhelet.
*In my humble opinion, although Mishna Brura rules that two out of four strings be Tekhelet, he leans more towards Rashi’s view on why two Tekhelet strings should be used and not Rabbeinu Tam’s view [see comment #11 of the Mishna Brura to O.C. chapter 12].
Also the Tur and the Chazon Ish (and in the view of the Beit Yosef {Rabbi Yosef Karo}, even the Raavad) stated that we should make the strings of Tekhelet in accordance to the amount mandated by Rashi and Tosafot {Rabbeinu Tam was one of the Tosafot} and so too does the author of Aruch Hashulchan {Rabbi Epstein} lean (but not in an absolute way). However, I must admit the truth, that currently, from what I see on the street, the majority (who are in actual way using Tekhelet) seem to be going by Meiri’s viewpoint and in accordance to the textual version of Raavad’s critiques of Rambam that we have (namely 2 out of 8). And some important Rabbis (for example, Rabbi Dov Lior, Shlit”a) act in accordance with Rambam’s viewpoint.
Rabbi Eliyahu Tevger, Shlit”a wrote:
On the question of the position of our Rabbi, the Gaon of Vilna, on this issue, we seemingly have a great confusion. In three places our Rabbi relates to the subject, regarding the number of strings of Tekhelet, and in each one, he expresses a different opinion. In his commentary to Shulchan Aruch (O.C. 12:5) he holds like the view of Rashi and Tosafot and shows support for their views from various sections of the Gemara and Sifri. In his commentary to Safra Dtsniuta (chapter 5, on the words Shiva Rehitin) he explains in accordance to Rambam, and in his commentary to the Zohar, Yahel Ohr (Parshat Pinchas 228b), he explains according to Raavad’s opinion {comment: that in his critiques of the Rambam, it is written in the texts that we have, that we use 2 Tekhelet strings out of the 8) and he writes that viewpoint that should be accepted is as he {Raavad} says.
For Rabbi Eliyahu Tevger’s explanation for the contradiction within the words of the Vilna Gaon, see the internet article in Hebrew at http://tekhelet.com/pdf/hagra.pdf . But in my humble opinion, there is another way to explain the words of the Vilna Gaon.
In the commentary of Tifferet Yisrael to Menachot Chapter 4, the author of Tifferet Yisrael establishes:
“That it is a mitzva to put on the each corner 4 strings. One or two strings of Tekhelet and the rest white. But if all of them were white or all of them were Tekhelet, he has fulfilled his obligation”.
Now it appears to me that in the opinion of Tifferet Yisrael there is an option either to use one Tekhelet string or two. For in his opinion, from the verse “G’dillim {a Hebrew term defined by Rabbi Aryeh Kaplan as, bound or doubled tassels} you shall make for yourself”, we learn just that there is an obligation to put 4 strings (before tying into 8) on every corner. But we don’t learn from this verse at all, how many strings out of the 4 have to be Tekhelet.
And if so it is possible to fulfill the obligation of the verse “and the corner is a P’til {a Hebrew term to be defined later on}of Tekhelet”, either by one string of Tekhelet or by two.
Now many ask, how is it possible that P’til of Tekhelet be more than one string, for seemingly, there would have been a need to write P’tilei {which denotes the plural form of P’til} if the intent was for two?
Now one can respond that this is an ancient dispute. For Beit Hillel and seemingly Targum Unkelos to Numbers 15:38 indeed held that P’til= a string. However, Beit Shammai (and the halacha is in accordance with Beit Shammai on this matter) and the Targum Yerushalmi explain that P’til = a twisted ply as shown in the attached picture (and does not instruct us on the amount of Tekhelet strings that we should use).
 | I partially unravelled a Tekhelet string for this picture to show that each Tekhelet string is a twisted ply. |
And so too, Rashi in his commentary to the Torah (Exodus 39:31) proves that the term P’til Tekhelet is not limited specifically to one string. Now here is a translated quote:
“And do not be surprised that it didn’t say {in the text that discusses the strings that fasten certain garments of the Temple priesthood} P’tilei Tekhelet {which more simply denotes multiple strings} since they are multiple {strings}, for behold we have found regarding the breastplate and the Ephod –‘They laced the breastplate … with a P’til Tekhelet‘ (Exodus 39:21) and you are compelled to agree that there were not less than two, for behold at the two edges of the breastplate were two rings of the breastplate and on the two shoulders of the Ephod, that were opposite them, and according to the way of tying, there were four strings, but in any case less than two is not possible”, {end of quote.}
One should further add that also the Hebrew word Oto {or Oso depending on your accent}spelled אותו
that appears in the section describing the mitzvah of Tzitzit (Numbers 15:39)
And it shall be for you for Tzitzit and you shall see אותו
where the word Oto is either defined in English as the word, it or as the word, him, although it is in the singular form, still is not a proof that one should use only one string of Tekhelet, for our Sages used this word to derive the Torah’s viewpoint on other issues regarding Tzitzit {not connected to the issue of how many Tekhelet strings to use}.
Therefore it appears to me, that as far as the absolute truth is concerned, the Vilna Gaon held that it is sufficient to use one string out of the four {before tying} for Tekhelet, however because there is an option similar to the viewpoint of the Tifferet Yisrael to use either one or two, for practical halachic action, we are obligated to be stringent in accordance to the views of Rashi and Tosafot (who demand that two out of the four strings be Tekhelet).