Structure conjecture for varieties with semi-positive holomorphic sectional curvature

Let $X$ be a smooth projective variety with semi-positive holomorphic sectional curvature. A smooth morphism is a morphism whose fibers are smooth, and a variety is rationally connected if two general points can be joined by a rational curve. A finite étale cover is a finite étale surjective morphism.

Structure conjecture. There exists a smooth morphism $X\to Y$ such that a fiber is rationally connected and $Y$ admits a finite étale cover $A\to Y$ by an abelian variety $A$. Moreover, the fiber product

$$\widetilde{X}:=X\times_Y A$$

is isomorphic to the product $A\times R$, where $R$ is a rationally connected projective variety.

The conjecture predicts a product decomposition after passing to a finite étale cover, analogous to the known structure results for semi-negative holomorphic sectional curvature. The source presents it as an open problem and notes that it was suggested by Junyan Cao.