The coprimality conjecture for reductions of two elliptic curves

Let $E_1$ and $E_2$ be non-CM elliptic curves over $\mathbb{Q}$. For $x\geq 1$, define

$$\pi_{E_1,E_2}^{\mathrm{coprime}}(x)=\#\{p\leq x:p\nmid N_{E_1}N_{E_2}\text{ and }\gcd(\#E_1(\mathbb{F}_p),\#E_2(\mathbb{F}_p))=1\},$$

where $N_{E_i}$ denotes the conductor of $E_i$ for $i=1,2$. Coprimality conjecture. As $x\to\infty$,

$$\pi_{E_1,E_2}^{\mathrm{coprime}}(x)\sim C_{E_1,E_2}^{\mathrm{coprime}}\cdot\frac{x}{\log x},$$

where $C_{E_1,E_2}^{\mathrm{coprime}}\geq 0$ is an explicit constant defined by the inclusion-exclusion model attached to the adelic Galois image of $E_1\times E_2$. This conjecture predicts the asymptotic density of primes at which the two reduction groups have coprime orders, extending the study of prime-factor conditions on elliptic-curve reductions; the explicit constant is intended to account for entanglements among the division fields of the two curves.