Grauert–Riemenschneider vanishing conjecture over simple normal crossing pairs

Let $Y$ be a normal excellent scheme, $D$ a Weil ${\mathbb Z}$-divisor on $Y$, and $\Delta_D$ a Weil ${\mathbb R}$-divisor whose coefficients are in $[0,1]$. Assume that $D+\Delta_D$ is ${\mathbb R}$-Cartier. Let $\pi\colon X\to Y$ be a proper, birational log resolution of $(Y,D+\Delta_D)$, and let $\pi^{\rm KF}(D)$ denote the K-twisted Fujita transform. Suppose in addition that $(Y,D+\Delta_D)$ is a simple normal crossing pair.

Grauert–Riemenschneider vanishing conjecture. For every $i>0$,

$$R^i\pi_*{\mathcal O}_X\bigl(\pi^{\rm KF}(D)\bigr)=0.$$

The paper proves this conjecture under the stated resolution assumption, and consequently obtains it for excellent $\mathbb{Q}$-schemes and excellent schemes of dimension at most $3$.