The Wall–Sun–Sun prime conjecture for Fibonacci numbers

Let $F_m$ denote the $m$th Fibonacci number, and let $\left(\frac{p}{5}\right)$ be the Legendre symbol for a prime number $p$. Wall–Sun–Sun prime conjecture. There are no prime numbers $p$ such that

$$p^2\mid F_{p-\left(\frac{p}{5}\right)}.$$

Such primes are also called Fibonacci-Wieferich primes. The conjecture is used in the paper to deduce square-freeness results for odd integers dividing sums of initial Fibonacci numbers, but its resolution is not established in the supplied text.