Klingler–Lerer conjecture on arithmetic points of strata

Let $\Omega\mathcal{M}_g(\kappa)$ be a stratum of abelian differentials, and let $S\subset\Omega\mathcal{M}_g(\kappa)$ be an irreducible algebraic subvariety. Call a subvariety $\overline{\mathbb{Q}}$-bi-algebraic if it is algebraic in period coordinates and, for both structures, defined over $\overline{\mathbb{Q}}$. Klingler–Lerer conjecture. If $S$ contains a Zariski-dense set of arithmetic points, then $S$ is $\overline{\mathbb{Q}}$-bi-algebraic. This is an André–Oort-inspired arithmeticity prediction for strata of abelian differentials; the supplied text gives no resolution status.