The Jacobi index-one sup-norm conjecture

Let $J^{cusp}_{k,1}(N)$ denote the space of Jacobi cusp forms of index $1$ and level $N$, and let $\sup(J^{cusp}_{k,1}(N))$ denote its associated supremum quantity. Jacobi index-one sup-norm conjecture. Let $k \ge 4$ be even. Then for large $N$, one has

$$\sup( J^{cusp}_{k,1}(N)) \asymp 1.$$

This would give the expected uniform bound for the size of the space of Jacobi cusp forms of index $1$; the paper notes that the available result only establishes bounds between $1$ and $N^{1/2}$, so the conjecture remains open.