Characterization of nontrivial common divisors of Pell-Lucas terms and indices

Let $(Q_n)_{n\geq 0}$ be the associated Pell sequence. For a prime $p$, let the rank of apparition (or entry point) $e_Q(p)$ be the smallest positive integer $r$ such that $p$ divides $Q_r$. The nontrivial-gcd conjecture. For every $k\geq 1$, one has

$$\gcd(Q_k,k)>1$$

if and only if there exists a prime $p$ such that $p$ divides $k$ and $e_Q(p)$ divides $k$. For example, $\gcd(Q_{21},21)=7$, and for $p=7$, both $7$ divides $21$ and $e_Q(7)=3$ divides $21$. This would characterize exactly when an associated Pell number and its index have a nontrivial common divisor; the supplied text gives the criterion as a claim and illustrates it, but provides no resolution beyond that example.