Conjecture on simultaneous supersingular reduction for elliptic-curve pairs

Let $E$ and $E'$ be two elliptic curves defined over a number field $K$. Assume that $E$ and $E'$ are non-CM and non-isogenous over $\overline{\mathbb{Q}}$. Let

$$\pi_{E,E',K}(x):=\left|\left\{p\leq x:\exists v\in\Sigma_{\operatorname{ssing}}(E,E')\text{ with }v\mid p\right\}\right|,$$

where $\Sigma_{\operatorname{ssing}}(E,E')$ is the set of finite places at which both curves have supersingular reduction. Simultaneous supersingular reduction conjecture. There exists a constant $C(E,E',K)\geq 0$ such that

$$\pi_{E,E',K}(x)\leq C(E,E',K)\log\log x$$

as $x\to\infty$.

This is a number-field analogue of the Lang–Trotter heuristic for pairs and would provide the sparsity estimate needed in the paper's height-bound argument. The source presents it as a natural conjecture and gives no resolution evidence.