The SKT metric conjecture on Frölicher spectral sequence degeneration

Let $X$ be a compact complex manifold. An SKT metric on $X$ is a C^ positive definite $(1,1)$-form $\omega$ satisfying

$$\partial\bar\partial\omega=0.$$

SKT degeneration conjecture. If $X$ admits an SKT metric, then its Frölicher spectral sequence degenerates at the second page $E_2$.

This conjecture connects the existence of special Hermitian metrics with Hodge-theoretic degeneration. It was proposed in earlier work and solved in a special case, while the general assertion remains open.