The Hodge filtration criterion for rational singularities of local complete intersections

Let $X$ be a smooth variety and let $Z\subseteq X$ be a local complete intersection of pure codimension $r$. Write $F_1\mathcal{H}_Z^r(\mathscr{O}_X)$ for the first Hodge-filtration piece and $E_1\mathcal{H}_Z^r(\mathscr{O}_X)$ for the corresponding first piece of the pole-order filtration. Rational-singularity conjecture. If

$$F_1\mathcal{H}_Z^r(\mathscr{O}_X)=E_1\mathcal{H}_Z^r(\mathscr{O}_X),$$

then $Z$ has rational singularities. This is proposed by analogy with the corresponding criterion for reduced hypersurfaces; whether the equality characterizes rational singularities in this local complete-intersection setting remains open.