Subharmonic variation conjecture for Azukawa pseudometrics of balanced domains

Let $\widetilde{\Omega} \subset \mathbb{C}^n \times \Delta$ be a pseudoconvex domain. For each $t \in \Delta$, define the fiber

$$\Omega_t=\{z \in \mathbb{C}^n:(z,t)\in\widetilde{\Omega}\}.$$

Fix $w\in\mathbb{C}^n$, and assume that every $\Omega_t$ is hyperconvex and contains $w$. Write $A_t(X):=A_{\Omega_t,w}(X)$ for the Azukawa pseudometric of $\Omega_t$ at $w$. For each $t$, set

$$I_t=\{X:A_t(X)<0\},$$

and let $V(t)$ denote the volume of $I_t$. Subharmonic variation conjecture. The function $(X,t)\mapsto A_t(X)$ is plurisubharmonic on $\mathbb{C}^n\times\Delta$, and the function

$$t\longmapsto-\log V(t)$$

is subharmonic on $\Delta$. This is the proposed pluripotential analogue of the subharmonic variation of Robin constants under pseudoconvex domain variations. The conjectural assertions concern both joint plurisubharmonicity of the Azukawa pseudometrics and subharmonicity of the logarithmic volume of their sublevel sets.