From “Breslov Eikh Shehu: Breslov Customs and Practices” by Dovid Zeitlin and Dovid Sears, work-in-progress
For more than a century, Klal Yisrael has witnessed the attempt to reinstate the mitzvah of tekheles, the blue thread that the Torah commands men to wear as part of the tzitsis (knotted strings) on each corner of a four-cornered garment. And for almost as many years, people have associated the renewal of tekheles with the Breslover Chassidim, among several other groups. If one sees a chassid wearing a talis with tekheles today, the odds are better than 50/50 that he is a Breslover. However, some may be surprised to learn that there has never been a consensus on this issue among the leaders of the Breslov community.
During the late 1800s, Rabbi Gershon Henich Leiner (d. 1890), the Radzyner Rebbe, attempted to restore this mitzvah, based on his extensive research into the identity of the khillazon—a sea-creature used for the precious blue dye. He presented his argument in several works, identifying the cuttlefish as the khillazon, and commissioned the production of what he believed to be the tekheles-strings, which his followers began to distribute.
(For more on this subject, see http://www.begedivri.com/techelet/Sefunei.htm.)
Reb Avraham b’Reb Nachman Chazan (d. 1919), author of Biur ha-Likkutim on Likutey Moharan and other Breslov seforim, met with the Radzyner Rebbe during the last years of his life and accepted the Radzyner tekheles. Some say that he did so based on secret traditions from the Rebbe's Megillas Sesarim ("Hidden Scroll") (see Rabbi Levi Yitzchak Bender, Si'ach Sarfei Kodesh III, 559). However, attempts thus far to decode the Megillas Sesarim have not substantiated this.
Breslover scholar Rabbi Nachman Goldstein (d. 1894), the Tcheriner Rov, who was an older contemporary of Reb Avraham b’Reb Nachman, took the opposing view. He argued that the Rebbe states in Likkutei Moharan I, 7, that tzitzis today are white (Parpara’os le-Chokhmah, ad loc.). Reb Noson also seems to support this in Likutey Halakhos, where he relates the wearing of tekheles to the Final Redemption (see Likutey Halakhos, Netilas Yadayim Shacharis 4:12; ibid. Tzitzis 1:1; ibid. Tzitzis 5:7, among other sources there.)
The Tcheriner Rov’s grandson and great-grandson of Reb Noson, Reb Avraham Sternhartz (d. 1955), also rejected wearing tekheles at the present time, as did another prominent descendant of Reb Noson, Reb Michel Sternhartz (son of Reb Yitzchak, to whom most of the letters in Alim le-Terufah are written). (Cf. Si'ach Sarfei Kodesh III, 338).
Therefore, some Breslover Chassidim of the early 20th century followed the view of Reb Avraham b’Reb Nachman and wore the Radzyner tekheles, while others followed that of the Tcheriner Rov and did not.
The restoration of the mitzvah of tekheles is not only a matter of halakhic dispute, but also contains mystical ramifications. Rabbi Gedaliah Kenig said in the name of his teacher Reb Avraham Sternhartz that the premature restoration of this custom could bring harsh judgments upon the world, since the blue thread is kabbalistically bound up with the forces of din.
(Heard from Rabbi Elazar Kenig. Possible support for this view may be found in Likutey Moharan I, 49:7, citing Zohar III, 152a).
Moreover, Reb Gedaliah did not believe the Radzyner tekheles to be authentic; but he contended that “even if we had the true tekheles, it would still be a dangerous”—for spiritual reasons.
(Heard from Rabbi Dovid Shapiro)
Yet Reb Gedaliah once said that if one possessed impeccable yiras Shomayim—fear of Heaven—he could wear techeiles. (This remark also suggests why the mitzvah of tekheles may have been lost.)
(Heard from Rabbi Elazar Kenig)
Rav Yaakov Herzog (1889-1959), later Chief Rabbi of Israel, eventually disproved the Radzyner Rebbe’s identification of the cuttlefish as the khillazon, and he initially believed the source of the precious dye to be the Janthina. However, subsequent research (in the course of which he may have been misinformed) led Rav Herzog to change his mind in favor of the murex trunculus. This research remained inconclusive until 1988, when Rabbi Eliyahu Tavger succeeded in producing a blue dye from the murex trunculus that seemed to meet all of the criteria of tekheles. Several years later, the Ptil Tekhelet Foundation began to make this dye available to the public and to argue for its acceptance. (For more on this, see the Ptil Tekhelet Foundation website, http://www.techeilet.co.il.)
Their efforts have meet with increasing success, and a number of prominent rabbis have begun to wear this tekheles (see http://www.tekhelet.com/mmlib.xml).
Yet some scholars, including Breslov Rabbi Dovid Shapiro, maintain that the Janthina is the khillazon and reject the arguments of the Ptil Tekhelet Foundation.
There is no consensus on any of this in the Breslov Chassidic community. Certain prominent figures such as Rabbi Eliezer Berland and Rabbi Eliezer Shlomo Schick have accepted the restoration of tekheles, while others such as Rabbi Elazar Mordechai Kenig, Rabbi Shmuel Moshe Kramer, Rabbi Nachman Burstein, and Rabbi Chaim Kramer have taken a more conservative view. However, if support for the “new tekheles” increases and the element of doubt were to be removed, a greater consensus would be reached within the Breslov community, as it would within the larger Jewish religious world. Until then, we remain in the category of “chalukas ha-eitzah,” uncertain about what to do (Likutey Moharan I, 61).