Ueno's Conjecture K strengthening for algebraic fiber spaces over abelian varieties

Let $f \colon X \to A$ be an algebraic fiber space, with $X$ a smooth projective variety and $A$ an abelian variety, and let $F$ be its general fiber. Assume that $f$ is smooth away from a closed set of codimension at least $2$ in $A$.

Ueno's Conjecture K strengthening. There exists an isogeny $A' \to A$ such that

$$X\times_{A} A' \sim A' \times F,$$

i.e. $X$ becomes birational to a product after an étale base change.

This conjecture strengthens the expected conclusion that such a fibration is birationally isotrivial. The source presents it as an analogy with Ueno's Conjecture K; the supplied text gives no resolution, so its status remains open.