Böcherer's conjecture for degree-two Siegel cusp forms

Let $k$) be even, let $f\in S_k(\operatorname{Sp}(4,\mathbb Z))$ be a non-zero Hecke eigenform, and let $\pi_f$ be the associated automorphic representation of $\operatorname{GSp}(4,\mathbb A)$. For an imaginary quadratic field $K=\mathbb Q(\sqrt d)$ with $d<0$ a fundamental discriminant, let $R(f,K)$ be the sum of the Fourier coefficients of $f$ over the ideal class group of $K$; write $w(K)$ for the number of roots of unity in $K$, $\chi_d$ for the associated quadratic Hecke character, and $L_{\mathrm f}(s,\pi_f\otimes\chi_d)$ for the finite part of the degree-$4$ Langlands $L$-function.

Böcherer's conjecture. There exists a constant $c_f$ depending only on $f$ such that, for every such $K$,

$$|R(f,K)|^2=c_f\,w(K)^2|d|^{k-1}L_{\mathrm f}(1/2,\pi_f\otimes\chi_d).$$

This conjecture relates class-group averages of Fourier coefficients to central $L$-values and is a fundamental instance of the connection between Siegel modular forms and automorphic $L$-functions. The source gives no resolution status for the conjecture.