The equivalence conjecture for Kähler and sG hyperbolicity on compact complex surfaces

Let $X$ be a compact complex surface. Kähler hyperbolicity and sG hyperbolicity are properties of $X$.

Kähler–sG hyperbolicity conjecture.

$$X\text{ is Kähler hyperbolic}\Longleftrightarrow X\text{ is sG hyperbolic}.$$

This conjecture concerns the relation between two hyperbolicity notions in complex dimension two and is motivated by deformation results for Kähler hyperbolic manifolds. Its resolution is not given in the source.