Mustață–Popa depth conjecture for the Du Bois complex

Let $X$ be a variety, let $i$ be a nonnegative integer, and write $\underline{\Omega}^{i}_{X}$ for the $i$-th graded piece of the Du Bois complex of $X$. For a sheaf or complex, let $\depth$ denote its depth, and let $\mathcal O_X$ be the structure sheaf. Mustață–Popa's depth conjecture. If

$$\depth(\mathcal O_X)\geq i+2,$$

then

$$\depth(\underline{\Omega}^{i}_{X})\geq 2.$$

This conjecture predicts that sufficiently deep singularities impose a uniform depth bound on every graded piece of the Du Bois complex. The paper presents the stated result as a partial answer to a conjecture of Mustață and Popa; its general status is not established here.