Positive common-divisor growth conjecture in characteristic p

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, let $E/\mathbb{F}_q(T)$ be an elliptic curve, and let $P,Q\in E(\mathbb{F}_q(T))$ be nontorsion points. Positive common-divisor growth conjecture. There is a constant $c=c(q,E,P,Q)>0$ such that

$$\deg\operatorname{GCD}(nP,nQ)\ge cn$$

for infinitely many $n\ge1$ with $p\nmid n$. This is proposed as the characteristic-$p$ analogue of the characteristic-zero common-divisor problem; the source does not claim a proof.