Level-aspect sup-norm conjecture for Siegel Hecke eigenforms

Let $F$ be as in the preceding individual-eigenform conjecture, let $\Gamma=\Gamma_0^{(n)}(N)$, and define

$$\lVert F\rVert_\infty:=\sup_{Z=X+iY\in\mathbf H_n}\det(Y)^{k/2}|F(Z)|.$$

Level-aspect sup-norm conjecture. With $F$ and $\Gamma$ as above, as $N\to\infty$,

$$\lVert F\rVert_\infty\ll_k N^{-n(n+1)/4}.$$

This is the expected individual-eigenform bound in the level aspect for $\Gamma_0^{(n)}(N)$. It is presented as a conjectural strengthening beyond what follows from the average Bergman-kernel estimates and remains open in the stated generality.