Normalized volume comparison for non-closed points

Let $(X,D)$ be a klt pair, let $\eta$ be a non-closed point, and let $Z=\overline{\{\eta\}}$ have dimension $d$. Choose a general closed point $x\in Z$, and let $n=\dim X$. Non-closed-point comparison conjecture.

$$\widehat{\operatorname{vol}}(x,X,D)=\widehat{\operatorname{vol}}(\eta,X,D)\cdot \frac{n^n}{(n-d)^{n-d}}.$$

The conjecture asserts that, after the stated scaling, non-closed points give no additional information beyond general closed points of their closure.