The Saito-Kurokawa newform sup-norm conjecture

Let $\mathrm{SK}_k(N)$ denote the space of Saito-Kurokawa lifts of weight $k$ and level $N$, and let $F\in\mathrm{SK}_k(N)$ be an $L^2$-normalized newform. Write $\lVert F\rVert_\infty$ for its supremum norm. Saito-Kurokawa newform sup-norm conjecture. Let $k\ge 4$ be even and $N\ge 1$. For an $L^2$-normalized newform $F\in\mathrm{SK}_k(N)$ one has

$$\lVert F\rVert_\infty\ll N^{-3/2}.$$

This is proposed as the individual-newform counterpart to the conjectured size of the full Saito-Kurokawa lift space; the paper does not establish this bound in general, so it remains open.