Baily–Borel–Hodge compactification conjecture for Calabi–Yau type hypersurfaces

Let $(n,d)$ be a pair with

$$\frac{n+1}{d}=m+1\in \mathbb{Z}.$$

Let $M$ be the GIT moduli space of degree $d$ hypersurfaces in $\mathbb{P}^n$ with $m$-rational singularities, and let $M^{\rm BBH}$ denote its Hodge-theoretic compactification. Let

$$\Phi:\mathbb{P}^{\binom{n+d}{d}-1}\dashrightarrow M^{\rm BBH}$$

be the rational map induced by the period construction. Baily–Borel–Hodge compactification conjecture. The space $M$ admits the Hodge-theoretic compactification $M^{\rm BBH}$, and the rational map $\Phi$ is a regular morphism on the locus parameterizing $m$-Du Bois hypersurfaces. This conjecture proposes a Baily–Borel-type compactification for Calabi–Yau type hypersurfaces, analogous to the construction announced for polarized klt log Calabi–Yau pairs; the paper expects a reduction to this statement after an appropriate cyclic cover, but no resolution is given here.