Shiffman–Zelditch's finiteness conjecture for Calabi volume of Bergman spaces

Let $M$ be a projective manifold, let $L$ be a very ample line bundle on $M$, and for each $k\in\mathbb{N}$ let $\mathcal{B}_{M,L^k}$ be the corresponding Bergman space with Calabi volume measure $\mu_{Ca}$. Shiffman–Zelditch's finiteness conjecture. The Calabi volume

$$\mu_{Ca}(\mathcal{B}_{M,L^k})$$

is finite for each $k$. The conjecture concerns the total volume of the finite-dimensional Bergman space equipped with the metric induced by the Calabi metric. The paper exhibits polarized manifolds with non-compact automorphism groups for which this volume is infinite, so the conjecture as stated is refuted.