Let $X$ be a projective variety with Du Bois singularities, which is additionally assumed to be Cohen-Macaulay. Then $H^i(X, \mathscr L^{-1}) = 0$ for any ample line bundle $\mathscr L$ and $i < \dim X$, by Thm. 10.42 of Kollár, Singularities of the MMP.

Question: Does this still hold if $\mathscr L$ is only big and nef?