Mustață–Popa's local cohomological vanishing conjecture

Suppose $X$ is embedded in a smooth variety $Y$ of dimension $d$. Choose a log resolution

$$f:Z\rightarrow Y$$

of the pair $(Y,X)$ that is an isomorphism away from $X$, and let

$$E=f^{-1}(X)_{\operatorname{red}}.$$

Thus $E$ is a simple normal crossings divisor on $Z$. Mustață–Popa's conjecture. If $\operatorname{depth} \mathscr{O}_X\geq j+2$, then

$$R^{d-2}f_*\Omega_Y^{d-j}(\log E)=0.$$

The conjecture relates depth of the structure sheaf to vanishing in the Hodge filtration on local cohomology. The supplied source does not state whether it has been resolved, so its status is recorded as open here.