The Bergman-kernel growth conjecture for Siegel cusp forms

Let $\mathcal H_n$ be the Siegel upper half-space of degree $n$, let $S_k^n$ be the space of Siegel cusp forms of degree $n$ and weight $k$, and let $\mathcal B_k^n$ be an orthonormal basis of $S_k^n$. For $Z=X+iY\in\mathcal H_n$, define

$$\mathbb B_k(Z,Z):=\sum_{F\in\mathcal B_k^n}|F(Z)|^2\det(Y)^k.$$

Bergman-kernel growth conjecture. With this notation,

$$\sup_{Z\in\mathcal H_n}\mathbb B_k(Z,Z)\asymp_n k^{3n(n+1)/4}.$$

The Bergman kernel is an invariant reproducing-kernel quantity whose growth controls aggregate pointwise bounds for Siegel cusp forms. The source reports this as suggested by earlier work, but provides no resolution of the conjectured estimate.