Asymptotic polarity conjecture for weak Jacobi forms

For each positive integer $m$, let $J_{0,m}$ denote the space of weak Jacobi forms of weight $0$ and index $m$, and define

$$J^P_{0,m}:=\{\varphi_{0,m}\in J_{0,m}\mid c(n,l)=0\text{ for }l^2-4mn>P\}.$$

Let $P(m)$ be the integer such that $J^{P(m)}_{0,m}=0$ and $J^{P(m)+1}_{0,m}\ne0$. Thus $P(m)$ measures the threshold for the smallest attainable maximal polarity of a nonzero weak Jacobi form. Asymptotic polarity conjecture. As $m\to\infty$,

$$P(m)=\frac{m}{2}+O(m^{1/2}).$$

This asymptotic behavior was conjectured by Gaberdiel and collaborators and is supported by computations of $P(m)$ for indices up to $61$ in the paper. Its general proof remains open.