The Saito-Kurokawa lift space-size conjecture

Let $\mathrm{SK}_k(N)$ denote the space of Saito-Kurokawa lifts of weight $k$ and level $N$, let $B_k^{SK}(N)$ be an orthonormal basis, and define

$$\mathbb B_N(Z)=\sum_{F\in B_k^{SK}(N)}\det(Y)^k|F(Z)|^2,$$

with $\sup(\mathrm{SK}_k(N))=\sup_{Z\in\mathbf H_2}\mathbb B_N(Z)$. Saito-Kurokawa lift space-size conjecture. For all even $k\ge 4$ and $N\ge 1$, one has

$$\sup(\mathrm{SK}_k(N))\asymp k^{5/2}N^{-2}.$$

At level $1$, the corresponding weight-aspect size is known to be of order $k^{5/2}$, while the paper proves only partial level-aspect bounds for prime levels. The conjecture predicts the additional factor $N^{-2}$ and remains open.