The volume formula for slices of extended infinitesimal Okounkov bodies

Let $X$ be a smooth projective variety of dimension $n$, let $\pi:{\rm Bl}_{r}(X)\rightarrow X$ be the blow-up at $r$ general points $x_{1},\dots,x_{r}$, and let $D$ be a big $\mathbb{R}$-divisor on $X$. For infinitesimal flags $Y^{1}_{\bullet},\dots,Y^{r}_{\bullet}$ over these points and ${\bf m}=(m_{1},\dots,m_{r})\in\mathbb{N}^{r}$, define the slice $S_{(m_{1},\dots,m_{r})}$ by the equations $\nu_{i}^{(1)}/m_{1}=\cdots=\nu_{i}^{(r)}/m_{r}$ for $i=1,\dots,n$. The volume formula conjecture. For all such flags and all ${\bf m}\in\mathbb{N}^{r}$, identifying $\mathbb{R}^{n}$ with $S_{(m_{1},\dots,m_{r})}$, one has

$${\rm vol}_{\mathbb{R}^{n}}\left(\left.\widetilde{\Delta}_{Y^{1}_{\bullet},\dots,Y^{r}_{\bullet}}(D)\right|_{S_{(m_{1},\dots,m_{r})}}\right)=\frac{(\sqrt{r})^{n-2}}{n!}{\rm vol}_{X}(D).$$

This conjecture concerns the intersection-theoretic information encoded by volumes of slices of extended Okounkov bodies; the source states that only a partial answer is obtained.