The product decomposition conjecture for the kappa-zero case

Let $X$ satisfy the assumptions that its cotangent bundle contains a nef locally free subsheaf $L^*\subset\Omega_X^1$ proportional to $K_X$. If $\kappa(X)=0$, then the product decomposition conjecture asserts that $K_X\equiv0$ and that there is a finite étale cover $\gamma:\widetilde X\to X$ such that $\gamma^*(L^*)=\mathcal O_{\widetilde X}$ and $\widetilde X=A\times Y$, where $A$ is Abelian and $Y$ is simply connected. The paper proves this conjecture in several cases, including $\dim X\leq4$, but leaves it open in general.