Wang's Hodge-theoretic criterion for finite Weil–Petersson distance

Let $\mathfrak{X}/S$ be an $n$-dimensional Calabi–Yau degeneration that is smooth outside a simple normal crossing divisor $\bigcup_i E_i$. For $s\in\bigcup_i E_i$, write $X:=\mathfrak{X}_s$. Let $F^n_\infty(s)$ be the $n$-th piece of Schmid's limiting Hodge filtration, and let $N_i$ be the nilpotent part of monodromy around $E_i$.

Wang's conjecture. $X$ has finite Weil–Petersson distance if and only if

$$N_iF^n_\infty(s)=0$$

for every $i$ such that $s\in E_i$.

This extends the one-parameter Hodge-theoretic criterion for finite Weil–Petersson distance to degenerations over a higher-dimensional base. Its validity would imply diameter boundedness of the family along real curves, while the higher-dimensional case requires control of terms arising from degenerations of mixed Hodge structures on intersections of boundary divisors.