Is it true that if a Fibonacci number $F_{n}$ divides the product of two Fibonacci numbers, then it must divide at least one of them?

Is it true that for all $n \ne 1,2,6,12$, there exists a prime divisor $p$ of $F_{n}$ such that the entry point (first appearance as a divisor in the Fibonacci sequence) of $p$ is at position $n$?

I can see that the second statement implies the first (at least for $n \ne 1,2,6,12$).

Also, Carmichael's theorem says that for all $n \ne 1,2,6,12$, there exists a prime divisor $p$ of $F_{n}$ such that $p \nmid F_k$ for (positive integer) $k \lt n$, and the second statement is equivalent to a sort of counterpart going in the other direction: for all $n \ne 1,2,6,12$, there exists a prime divisor $p$ of $F_{n}$ such that $p \nmid F_k$ for $k \gt n$ save only for the obvious exceptions where $k$ is a multiple of $n$ (since $F_{n}\mid F_{k}$ when $n \mid k$).

These seem like natural questions to ask but I haven't seen them addressed.