Fujita-transform conjecture on birational invariance of higher direct images

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.

Fujita-transform conjecture. The derived pushforward

$$\mathbf{R}\pi_*{\mathcal O}_X\bigl(\pi^{\rm KF}(D)\bigr)$$

is independent of the choice of $X$.

It is suggested that this may follow from existing methods, while the paper proves it under a resolution assumption; in particular, the result holds for excellent $\mathbb{Q}$-schemes and excellent schemes of dimension at most $3$.