Q-abelianity conjecture for non-uniruled varieties with polarized endomorphisms

Let $X$ be a normal projective variety admitting a non-isomorphic polarized endomorphism. A normal projective variety is non-uniruled if it is not covered by rational curves, and it is Q-abelian if there is a finite surjective morphism from an abelian variety to it that is étale in codimension one. Q-abelianity conjecture. If $X$ is non-uniruled, then $X$ is Q-abelian.

The conjecture proposes that the weak Calabi–Yau factor in the structural theorem for polarized endomorphisms is always a point in the non-uniruled case. The surrounding theorem establishes that such an $X$ has Kodaira dimension zero and admits the relevant finite cover and decomposition, but the asserted Q-abelianity is not established here.