Nakayama's splitting conjecture for elliptic fibrations

Let $X$ be a normal projective variety of dimension $n-1$, and let $f:X\to X$ be an endomorphism of degree $d>1$. Suppose that $X$ admits a fibration $\tau:X\to Y$ whose general fibre is an elliptic curve and that the fibration commutes with $f$. Then, up to base change, $X$ is a product $Y\times E$, where $E$ is an elliptic curve.

Nakayama's splitting conjecture. Under these hypotheses, $X\simeq Y\times E$ after a base change.

The statement is proposed as the higher-dimensional analogue of a result due to Nakayama in the surface case and is intended to support the classification strategy for compact Kähler manifolds with $\kappa(X)=n-2$ and $a(X)=n-1$. The source does not establish the assertion in the stated generality.