Deng's isotriviality conjecture for families with semi-ample canonical bundle

Let $f:X\to Y$ be a smooth projective family between complex quasi-projective manifolds. Assume that each fiber of $f$ has semi-ample canonical bundle, and that the Kobayashi pseudodistance on $Y$ vanishes identically. Deng's isotriviality conjecture. Then $f$ is birationally isotrivial, or even isotrivial. This is an isotriviality analogue of higher-dimensional Shafarevich hyperbolicity: known results establish hyperbolicity for families with variation, while the source proposes this vanishing-pseudodistance criterion as a conjectural converse. The source gives no resolution status.