Deformation invariance of volumes of adjoint classes in smooth Kähler families

Let $f:X\to S$ be a smooth family from a Kähler manifold $X$ onto a smooth, connected, relatively compact curve $S$. Assume that $(X,B+\boldsymbol{\beta})$ is a generalized klt pair, $(X,B)$ is log smooth over $S$, and $\boldsymbol{\beta}=\overline{\beta}$ where $\beta$ is nef over $S$. Volume invariance conjecture. The function

$$t\longmapsto \operatorname{vol}(K_{X_t}+B_t+\boldsymbol{\beta}_{X_t})$$

is constant over $S$. The source presents this as a natural conjecture following local constancy results for the volume function; its general validity remains open.