Streets–Tian conjecture on closed Hermitian-symplectic complex manifolds

A Hermitian-symplectic metric on a complex manifold $(X,J)$ is a Hermitian metric whose associated fundamental $2$-form is the $(1,1)$-part of a symplectic form. A complex manifold is Hermitian-symplectic if it admits such a metric. The manifold is closed if it is compact without boundary.

Streets–Tian conjecture. Any closed Hermitian-symplectic complex manifold $(X,J)$ admits a Kähler metric.

This conjecture proposes that the Hermitian-symplectic condition on a closed complex manifold forces the existence of a Kähler metric. Its resolution status is not established by the supplied material.