On Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type

Given a Cohen-Macaulay scheme of klt type $X$ and a resolution $\pi \text{\hspace{0.17em}⁣}:Y\to X$, we show that $R^1{\pi }_{\ast }{\omega }_Y=0$. We deduce that if $\mathrm{d}\mathrm{i}\mathrm{m}(X)=3$, then $X$ satisfies Grauert-Riemenschneider vanishing and therefore has rational singularities. We also obtain that in arbitrary dimension, if $X$ is of finite type over a perfect field of characteristic $p>0$, then $X$ has ${\mathbb{Q}}_p$-rational singularities.