Blomer's sup-norm conjecture for Siegel Hecke eigenforms

Let $S_k^n$ denote the space of Siegel cusp forms of degree $n$ and weight $k$, and let $F\in S_k^n$ be an $L^2$-normalised Hecke eigenform. Write $\|F\|_\infty$ for its sup-norm. Blomer's conjecture. As $k\to\infty$, one has

$$\|F\|_\infty=k^{n(n+1)/8+o(1)}.$$

This conjecture concerns the expected size, in the weight aspect, of holomorphic Siegel Hecke eigenforms. The source attributes it to speculation by V. Blomer based on results for Saito--Kurokawa lifts; no resolution is given here.