Conjecture on Weil–Petersson metric completion for stable families

Let $O{X}\rightarrow O{S}$ be a stable family of smooth canonical models, with smooth locus $O{S}^\circ$ and Weil–Petersson metric $\omega_{WP}$ on $O{S}^\circ$. Assume that the family is nowhere infinitesimally trivial.

Weil–Petersson extension and completion conjecture. The metric $\omega_{WP}$ extends uniquely to a non-negative closed $(1,1)$-current on $O{S}$ with continuous local potentials, and the metric completion of $(O{S}^\circ,\omega_{WP})$ is homeomorphic to $O{S}$.

The conjecture extends the known bounded-potential and finite-distance results for special degenerations of canonical models. It predicts both regularity of the Weil–Petersson current across the boundary and a precise description of the metric completion.