Nonvanishing conjecture for Siegel Poincaré series of exponential type

Let $\Gamma_n=\operatorname{Sp}_n(\mathbf Z)$, let $T$ be a positive-definite index for the Siegel Poincaré series $P_{T,\Gamma_n}$, and assume the usual parity condition for scalar-weight Siegel modular forms. Nonvanishing conjecture for Siegel Poincaré series. For $k>n+1$ and $nk\equiv0\pmod 2$, all the Siegel Poincaré series $P_{T,\Gamma_n}$ of exponential type are non-zero.

Nonvanishing of these Poincaré series would give a general existence result for Siegel cusp forms with prescribed exponential-type Fourier data. The source describes this as an outstanding open question in the classical theory of modular forms.