Squarefree Fibonacci Numbers

I assume the traditional definition with $F_0=0$ and $F_1=1$.

Most likely there are infinitely many squarefree Fibonacci numbers. A simple way to construct them is to consider a subsequence $F_p$ for prime $p$. Notice that if $q^2\mid F_p$ for some prime $q$, then $q$ must be a Wall-Sun-Sun prime, whose existence is a big open question (and even if they exist, they would be very rare).

For prime $p$, let $M(p)$ be the least positive $n$ such that $p^2 \mid F_n$. Then $p^2 \mid F_n$ iff $M(p) \mid n$. Thus at most $N/M(p)$ of the first $N$ Fibonacci numbers are divisible by $p^2$. If we could prove that $\sum_p 1/M(p) < 1$, then at least a positive fraction of all Fibonacci numbers will be squarefree.
Well, it seems to be true numerically; I don't know if it's provable.