Uniform normalization conjecture for fiberwise plurisubharmonic functions

Let $f:X\to Z$ be as in the setting, with fibers $X_t$, forms $\theta_t$ and $\omega_t$, and let $V$ denote the common volume appearing in the fiberwise averages. Write $\operatorname{PSH}(X_t,\theta_t)$ for the set of $\theta_t$-plurisubharmonic functions on $X_t$. Uniform normalization conjecture. In the Setting, there exists a constant $C>0$ such that

$$\sup_{X_t}\varphi_t-C\leqslant \frac{1}{V}\int_{X_t}\varphi_t\,\omega_t^n\leqslant \sup_{X_t}\varphi_t$$

for all $t\in\overline{\mathbb D}_{1/2}$ and every function $\varphi_t\in\operatorname{PSH}(X_t,\theta_t)$. This conjectural uniform comparison extends the classical comparison between the supremum and mean value of plurisubharmonic functions from a fixed compact Kähler variety to a family, and supplies the uniform normalization needed in the family estimates.