The SKT-to-Kähler hyperbolicity conjecture

Let $X$ be a compact Kähler manifold, and let $\omega$ be a Kähler metric on $X$ that is SKT hyperbolic.

SKT-to-Kähler hyperbolicity conjecture. If $X$ admits such a metric $\omega$, then $X$ is Kähler hyperbolic.

The source explains that an $L^\infty$ solution of the relevant $\bar\partial$-problem would prove this implication. Thus the conjecture remains open in the stated generality.