Böcherer's paramodular conjecture for prime-level eigenforms

Let $p$) be prime and let $F\in S^k(\Gamma^{\mathrm{para}}[p])^+$ be a paramodular Hecke eigenform of even weight $k$. For each fundamental discriminant $D<0$, let $A_F(D)$ denote the normalized average of the Fourier coefficients of $F$ indexed by quadratic forms of discriminant $D$, and let $\chi_D$ be the associated quadratic character. Put $\alpha_D=1+\left(\frac{D}{p}\right)$. Böcherer's conjecture.

$$A_F(D)^2=\alpha_D C_F L(F,1/2,\chi_D)|D|^{k-1},$$

where $C_F$ is a nonnegative constant depending only on $F$. If $F$ is a Gritsenko lift, then $C_F>0$; if $F$ is not a lift, then $C_F=0$ if and only if $L(F,1/2)=0$. This is the paramodular analogue of Böcherer's conjecture relating quadratic-twist central values to averages of Fourier coefficients; it is proved for lifts and supported numerically for nonlifts.