Generalized common-divisor conjecture for elliptic curves over function fields

Let $K$ be a characteristic-zero function field, let $E_1/K$ and $E_2/K$ be elliptic curves, and let $P_1\in E_1(K)$ and $P_2\in E_2(K)$ be $K$-independent, meaning that no nonzero isogenies defined over $K$ identify a nontrivial multiple of one point with an isogenous image of the other. Define $\operatorname{GCD}(P_1,P_2)$ as the greatest common divisor of the pullbacks of the zero sections along the corresponding sections. Generalized common-divisor conjecture. There is a constant $c=c(K,E_1,E_2,P_1,P_2)$ such that

$$\deg\operatorname{GCD}(n_1P_1,n_2P_2)\le c\qquad\text{for all }n_1,n_2\ge1.$$

Furthermore,

$$\operatorname{GCD}(nP_1,nP_2)=\operatorname{GCD}(P_1,P_2)$$

for infinitely many $n\ge1$. This extends the same-curve, same-function-field formulation; the source presents it as a conjectural generalization and does not establish it in full.