Structure conjecture for compact LCK nilmanifolds

Let $M$ be a compact nilmanifold admitting a locally conformally Kähler (LCK) structure, and let $H(n)$ be the generalized Heisenberg group. Structure conjecture. Up to covering, $M$ is the product of the generalized Heisenberg group with $S^1$. This extends the known result for left-invariant LCK structures, for which the relevant nilmanifolds are of Vaisman type; the source states the broader claim as a belief rather than an established theorem.