I'll transcribe what Rav Nissan Kaplan said:
One reason (not to wear the murex techeiles) like Rav Elyashiv writes in the teshuva, that even if we know how to do the techeiles, there's like four or five shitos what has to be the techeiles (how many strings need to be dyed)1. Now...there's a raya...let me see if I remember the raya now. There's a raya in Bava Metzia 6b. I don't remember now what exactly the raya is, but I'm sure the lomdonim listeners here will be able to figure it out, there's a Ritva there. It's that if you mima nafshach don't know what shita to do, you don't have to do none (meaning, any).
Analyzing what he's saying, it sounds like the proof he brings from the Ritva isn't from Rav Elyashiv. Indeed, seeing the original teshuva from Rav Elysahiv (Kovetz Teshuvos I Orach Chaim § 2), he doesn't quote a Ritva. He instead cites the Yeshuos Malko Orach Chaim § 2,3 who says that one of the reasons not to wear techeiles today is there's a dispute between the Rambam and Rashi how to manufacture it (boil it with added chemicals or add the chemicals after boiling). Meaning, since we today can't rule how to make it, apparently that means there's no obligation to bother (as the Yeshuos Malko clarifies there, since one fulfills tzitzis even without techeiles. I imagine if that weren't true he would say differently).
Looking at the gemarra and Ritva there I was unable to discern what he was referring to. As he himself said, he doesn't remember what the proof was. Note that there are two Ritva's on Bava Metzia. There's the old print, which apparently is falsely attributed to the Ritva, and there's the new print which is meant to be correct. I'm not sure which one he was referring to, but I wasn't successful at finding anything when scanning through both.
1 Inspecting the teshuva it is clear Rav Nissan forgot that Rav Elyashiv wrote regarding that we don't know "how to do the techeiles", not about "how many strings". Doesn't make a difference though to the point at hand.