Classical-group descent conjecture for Whittaker-Fourier coefficients

Let $G$ be a quasi-split classical group, let $\psi_N$ be a fixed generic character, and let $\{\pi_1,\dots,\pi_k\}$ be distinct cuspidal representations of general linear groups satisfying the self-duality conditions required for descent. Let $\sigma^{\psi_N}(\{\pi_1,\dots,\pi_k\})$ be the resulting multiplicity-free $\psi_N$-generic cuspidal representation of $G(\mathbb A)$, and let $\theta$ be the outer involution occurring for $G=\operatorname{SO}(2n)$.

Classical-group descent conjecture. If $\pi$ is an irreducible constituent of $\sigma^{\psi_N}(\{\pi_1,\dots,\pi_k\})$, then

$$c_\pi^{\psi_N}=\begin{cases}2^{k-2}&\text{if $G=\operatorname{SO}(2n)$ and }\theta(\pi)=\pi,\\2^{k-1}&\text{otherwise.}\end{cases}$$

This is the concrete classical-group form of the general conjecture, using the descent construction and functorial transfer to general linear groups. Its validity is not established in the stated generality.