Akbary–Park conjecture on fixed traces for a pair of elliptic curves

Let $E_1$ and $E_2$ be non-isogenous elliptic curves over $\mathbb{Q}$, with conductors $N_1$ and $N_2$, respectively, and both without complex multiplication. For fixed integers $t_1$ and $t_2$, define

$$S(E_1,E_2,t_1,t_2;x)=\#\{p\leq x:p\nmid N_1N_2,\ a_p(E_1)=t_1\ \operatorname{and}\ a_p(E_2)=t_2\}.$$

Akbary–Park fixed-traces conjecture. There exists a constant $C(E_1,E_2,t_1,t_2)\geq0$ such that

$$S(E_1,E_2,t_1,t_2;x)\sim C(E_1,E_2,t_1,t_2)\operatorname{log}\operatorname{log}x$$

as $x\to\infty$. This is a more general conjecture about simultaneous prescribed Frobenius traces for two non-isogenous curves; setting $t_1=t_2=0$ recovers a precise form of the supersingular-pair conjecture. The source attributes it to Akbary and Park and focuses on the conjectural constant.