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

The online encyclopedia Wikipedia has the articles [*Fibonacci number*](https://en.wikipedia.org/wiki/Fibonacci_number) and [*Nontotient*](https://en.wikipedia.org/wiki/Nontotient), respectively.

I computed the initial 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.