Finiteness of the Calabi volume of the Bergman space

The Calabi metric on the space of Kähler metrics in a fixed Kähler class is the restriction of the deWitt–Ebin metric, and its finite-dimensional approximations are domains in finite-dimensional spheres. For each positive integer $k$, let $\mathcal{B}_k$ denote the corresponding finite-dimensional space, and let $G|_{\mathcal{B}_k}$ be the restriction of the metric $G$ to it. Finiteness conjecture. The Calabi volume

$$\operatorname{Vol}_k(\mathcal{B}_k)$$

with respect to $G|_{\mathcal{B}_k}$ is finite for each $k$. This asserts that the finite-dimensional approximation has finite volume, as suggested by the spherical geometry of the Calabi metric; the supplied text does not indicate whether the claim has been proved or remains open.