The isotriviality conjecture for fibrations with simple fibres

Let $f:X\to Y$ be a fibration, with $X$ and $Y$ smooth connected compact Kähler manifolds and $X$ in the class $C$. Write $X_y$ for a general fibre, and let $a(Y)$ denote the algebraic dimension of $Y$. The variation $Var(f)$ is zero when, after a proper connected base change $B\to Y$, the induced fibration is bimeromorphic over $B$ to a product $F\times B$.

Isotriviality conjecture for simple fibres. If $X_y$ is simple and $a(Y)=0$, then

$$Var(f)=0.$$

Equivalently, the general fibres of $f$ are pairwise bimeromorphic.

The conjecture predicts that fibrations over bases of algebraic dimension zero with simple general fibre have no bimeromorphic variation. The source notes related positive results for several classes of fibres, including Kummer and hyperkähler fibres, but does not specify a resolution of this general statement.