Bogomolov-Sommese vanishing for log canonical varieties

Let $(Z,\Delta)$ be a logarithmic pair, with $Z$ a variety and $\Delta$ a boundary divisor, and assume that $(Z,\Delta)$ is log canonical. Let $\Omega_Z^{[p]}(\log\Delta)$ denote the reflexive sheaf of logarithmic $p$-forms, and let $\mathscr A\subseteq\Omega_Z^{[p]}(\log\Delta)$ be a reflexive subsheaf of rank one. Assume that $\mathscr A$ is $\mathbb Q$-Cartier. Bogomolov-Sommese vanishing for log canonical varieties. Then

$$\kappa(\mathscr A)\leq p.$$

This is the expected extension of Bogomolov-Sommese vanishing from simple normal crossing pairs to log canonical pairs. It was verified under the additional assumption $\dim Z\leq 3$ in the cited work of Greb, Kebekus and Kovács, so the general statement remains open.