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

Let $n\geq 1$, let $\Gamma\subset\operatorname{Sp}_n(\mathbf Z)$ be a congruence subgroup of level $N$, and let $S^{(n)}_k(\Gamma)$ and $\mathbb B_{k,\Gamma}(Z)$ be as above. Level-aspect Bergman-kernel size conjecture. With $k$ fixed, as the level $N$ of $\Gamma$ tends to infinity,

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

This is the proposed correct $L^2$-size in the level aspect. The paper explains that lower bounds are available and that a hybrid asymptotic would be desirable, but the conjectured upper and lower comparison in the general non-compact arithmetic setting remains open.