Steenbrink's conjecture on Du Bois divisors

Let $X$ be a variety, let $D\subset X$ be a reduced Cartier divisor, and let $x\in D$. Assume that $D$ has only Du Bois singularities in a neighborhood of $x$. Steenbrink's conjecture. Then $X$ has only Du Bois singularities in a neighborhood of $x$. This is presented as the deformation question for Du Bois singularities and is attributed to Steenbrink; the source explains that the statement would imply the corresponding deformation result, while a weaker version with an additional hypothesis on $X\setminus D$ is the best known result given there.