This question is related to sequence of numbers $t$ such that $F_{6t}$ is a nontotient where $F_n$ represents the sequence of Fibonacci numbers for $n\geq 0$.

The online encyclopedia Wikipedia has the articles Fibonacci number and Nontotient, respectively.

There are very few terms in https://oeis.org/A335976 and some prime numbers appear as initial terms as expected.

Conjecture. There are infinitely many numbers $t$ such that $F_{6t}$ is a nontotient.

Question. Can someone prove or disprove above conjecture?

Initial terms of sequence of composite numbers $t_{c}$ such that $F_{6t_{c}}$ is a nontotient are also very welcome as helpful comment. Additionally, I couldn't find that question in literature yet, but if one can find references that have these or strongly related results, I will be very grateful for this response.

Thanks.