Birational isotriviality conjecture for smooth families over special bases

Let $f:X\to Y$ be a smooth projective morphism with connected fibres between two quasi-projective manifolds. Assume that $K_{X_y}$ is pseudo-effective for general $y\in Y$, and that $Y$ is special, meaning that it does not admit a dominant rational fibration whose orbifold base is of general type. Birational isotriviality conjecture. Then $f$ is birationally isotrivial. This would extend known results for families whose fibres have good minimal models or semi-ample canonical bundles, and would imply a complete solution of Conjecture 3.4 of Popa. The conjecture remains unresolved in the stated generality.