Injectivity conjecture for the intersection complexes of rational singularities

Let $X$ be a normal variety with pre-$(k-1)$-rational singularities. Let $I\underline{\Omega}_X^k$ denote the $k$th intersection complex, and let $\mathcal H^0 I\underline{\Omega}_X^k$ be its zeroth cohomology sheaf. The canonical morphism

$$\mathcal H^0 I\underline{\Omega}_X^k\to I\underline{\Omega}_X^k$$

induces, by dualizing, a natural morphism into the dualizing complex.

Intersection-complex injectivity conjecture. The natural morphism

$$\boldsymbol{R}\mathcal H om(I\underline{\Omega}_X^k,\omega_X^{\bullet})\to \boldsymbol{R}\mathcal H om(\mathcal H^0 I\underline{\Omega}_X^k,\omega_X^{\bullet})$$

is injective on cohomology.

This is the intersection-complex analogue of the main Du Bois injectivity conjecture. It is proposed for normal varieties with rational singularities in the relevant degrees, and the source does not give a general proof or resolution.