Lang–Trotter conjecture for simultaneous supersingular reduction

Let $(E_1,E_2)$ be a pair of elliptic curves defined over $\mathbb{Q}$. Assume that neither curve has complex multiplication and that they are not isogenous over $\overline{\mathbb{Q}}$. Define

$$\pi_{E_1,E_2}(x):=\#\left\{p\leq x:E_1\text{ and }E_2\text{ have supersingular reduction modulo }p\right\}.$$

Lang–Trotter conjecture for pairs. There exists a non-negative constant $c_{E_1,E_2}$ such that

$$\pi_{E_1,E_2}(x)\sim c_{E_1,E_2}\log\log x$$

as $x\to\infty$.

Lang and Trotter proposed this asymptotic as part of a probabilistic model for Frobenius traces, based on Sato–Tate heuristics. The source recalls it as an input for subsequent height bounds and gives no resolution status.