Decomposition conjecture for varieties with stable tangent sheaf and trivial canonical class

Let $X$ be a normal complex projective variety with klt singularities and $K_X \equiv 0$. Suppose that the tangent sheaf $T_X$ is stable with respect to some polarization. A finite morphism is quasi-étale if it is étale in codimension one.

Decomposition conjecture. There exists a quasi-étale cover $Y \to X$ such that either $Y$ is an abelian variety, or $Y$ splits as a product of copies of a single Calabi–Yau variety or of a single irreducible symplectic variety.

This conjecture gives a stronger factorization than the decomposition theorem established in the paper, which allows products of potentially different factors. The stated product structure is expected under a weak analogue of the Beauville–Bogomolov decomposition theorem; its general validity is not established here.