Positive-density annihilating-differential conjecture

Let $C$ be a genus-$2$ curve with Jacobian $J$, and let $\omega$ be an annihilating differential for the relevant Mordell–Weil subgroup. For each prime $p$ of good reduction, let $\omega_p$ be a suitably scaled annihilating differential and let $\bar{\omega}_p$ denote its reduction on $C(\mathbb F_p)$. Say that $J$ splits if it is isogenous over $\mathbb Q$ to a product of elliptic curves $E\times E'$. In the split case, let $\omega$ be the associated global annihilating differential.

Positive-density conjecture. The set of primes $p$ of good reduction for $C$ such that $\bar{\omega}_p$ does not vanish on $C(\mathbb F_p)$ has positive density, unless $J$ splits and $\omega$ vanishes at a rational point of $C$.

Such primes are needed to choose a modulus for the Mordell–Weil sieve combined with Chabauty. The source gives no resolution of this conjecture; the stated exception reflects the obstruction arising from a split Jacobian.