  Weak Jacobi forms of weight $0$ and index $m$ can be exponentially lifted to
meromorphic Siegel paramodular forms. It was recently observed that the Fourier
coefficients of such lifts are then either fast growing or slow growing. In
this note we investigate the space of weak Jacobi forms that lead to slow
growth. We provide analytic and numerical evidence for the conjecture that
there are such slow growing forms for any index $m$.
