Prove that $\forall p \in \Bbb P;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$

Prove that $\forall p \in \Bbb P,n \in \Bbb Z^+;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$ and $F_{5^n} \equiv 0 \mod 5^n$, where $\left(\dfrac{5}p\right)$ is the Legendre symbol and $F_m$ is the $m$th Fibonacci number, with $F_1=F_2=1$. This is a generalization of the identity $F_{p-\left(\frac{5}p\right)} \equiv 0 \mod p$. Is the generalization well-known? I was trying to find an upper bound on the index of the first Fibonacci number dividing any integer $m$, which would then be the product of the indices associated with the largest power of each prime dividing $m$.

It is known that if $p$ is an odd prime factor of $F_n$, then $\frac{F_{p n}}{F_n}$ is divisible by $p$ but not by $p^2$. For $p=2$, the situation has to be handled more carefully, but is also well known (cf. e.g. N.N. Vorobiev, Fibonacci Numbers, p. 76).