Rational ambient characterization conjecture for Du Bois singularities

Let $X$ be a variety, let $Y$ be an ambient variety, and suppose the criterion referred to as the Easy Du Bois Criterion characterizes Du Bois singularities when $Y$ is smooth. A variety has rational singularities when its higher direct images of the structure sheaf vanish on a resolution. Rational ambient characterization conjecture. The Easy Du Bois Criterion should remain true if the hypothesis that $Y$ is smooth is replaced by the condition that $Y$ has rational singularities. This would give a more natural characterization by weakening the smoothness assumption on the ambient variety; the source provides no resolution status.