Weight-aspect Bergman-kernel size conjecture for Siegel cusp forms

Let $n\geq 1$, let $\Gamma\subset\operatorname{Sp}_n(\mathbf Z)$ be cofinite, and let $S^{(n)}_k(\Gamma)$ be the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. If $B_k(\Gamma)$ is an orthonormal basis, define

$$\mathbb B_{k,\Gamma}(Z)=\sum_{F\in B_k(\Gamma)}\det(Y)^k|F(Z)|^2,\qquad Z=X+iY\in\mathbf H_n.$$

Weight-aspect Bergman-kernel size conjecture. As $k\to\infty$,

$$\sup_{Z\in\mathbf H_n}\mathbb B_{k,\Gamma}(Z)\asymp_{n,\Gamma} k^{3n(n+1)/4}.$$

This predicts the correct $L^2$-size of the space in the weight aspect. The corresponding lower bounds are accessible, and the expected size is motivated by the identity and parabolic contributions to the Bergman kernel, but the asserted asymptotic order in the non-compact arithmetic setting remains open.