Streets–Tian conjecture on Hermitian-symplectic compact complex manifolds

Let $M$ be a compact complex manifold. A Hermitian-symplectic metric on $M$ is a Hermitian metric whose Kähler form is the $(1,1)$-part of a closed $2$-form. Equivalently, there is a global $(2,0)$-form $\alpha$ such that

$$\Omega=\alpha+\omega+\overline{\alpha}$$

is closed, where $\omega$ is the Kähler form of the metric. Streets–Tian conjecture. If $M$ admits a Hermitian-symplectic metric, then it admits a Kähler metric. This is an important open question in non-Kähler geometry. The conjecture is known to be true in complex dimension $2$, but remains open in dimensions $3$ and higher.