Streets–Tian–Fino–Vezzoni intersection conjecture

Let $M$ be a compact complex manifold. A Hermitian-symplectic metric is a Hermitian metric whose Kähler form is the $(1,1)$-part of a closed $2$-form. A balanced metric is a Hermitian metric whose Kähler form $\omega$ satisfies $d(\omega^{n-1})=0$ on a complex manifold of dimension $n$.

Streets–Tian–Fino–Vezzoni intersection conjecture. If $M$ admits a Hermitian-symplectic metric and a balanced metric, then it must admit a Kähler metric.

This is the intersection of the Streets–Tian and Fino–Vezzoni conjectures. The paper states that it remains open to the authors' knowledge.