The metric-completion conjecture for LCK surfaces with proper potential

Let $M$ be a compact LCK complex surface with proper potential, and let $\widetilde M$ be its Kähler $\mathbb Z$-covering. The metric-completion conjecture. The metric completion of $\widetilde M$ is a Stein variety.

The statement is presented as a conjecture in the surrounding discussion because the proof is unavailable for class VII$_0$ non-Kato surfaces; the corresponding assertion is known for Vaisman surfaces.