Refined Böcherer conjecture for class-group characters

Let $f\in S_k(\operatorname{Sp}(4,\mathbb Z))$ be a non-zero Hecke eigenform of weight $k\geq2$, let $\pi_f$ be its associated automorphic representation of $\operatorname{GSp}(4,\mathbb A)$, and suppose that $f$ is not a Saito–Kurokawa lift. For an imaginary quadratic field $K=\mathbb Q(\sqrt d)$ with $d<0$ a fundamental discriminant and a character $\Lambda$ of $\operatorname{Cl}_K$, define $R(f,K,\Lambda)=\sum_{c\in\operatorname{Cl}_K}a(f,c)\Lambda^{-1}(c)$. Let ${\mathcal{AI}}(\Lambda^{-1})$ denote the automorphic induction of $\Lambda^{-1}$ to $\operatorname{GL}(2,\mathbb A)$, and let $w(K)$ be the number of roots of unity in $K$.

Refined Böcherer conjecture. For every such $K$ and $\Lambda$,

$$\frac{|R(f,K,\Lambda)|^2}{\langle f,f\rangle}=2^{2k-6}w(K)^2|d|^{k-1}\frac{L(1/2,\pi_f\times {\mathcal{AI}}(\Lambda^{-1}))}{L(1,\pi_f,\operatorname{Ad})}$$

and equivalently

$$\frac{|R(f,K,\Lambda)|^2}{\langle f,f\rangle}=\frac{2^{4k-6}\pi^{2k+1}}{(2k-2)!}w(K)^2|d|^{k-1}\frac{L_{\mathrm f}(1/2,\pi_f\times {\mathcal{AI}}(\Lambda^{-1}))}{L_{\mathrm f}(1,\pi_f,\operatorname{Ad})}.$$

This is a precise refinement of Böcherer's conjecture, specifying the constant through adjoint and Rankin–Selberg $L$-values. The source presents it as a conjectural extension; no resolution is supplied.