Descent multiplicity conjecture for classical and metaplectic groups

Let $G$ be a quasi-split classical group or a metaplectic group, and let $\sigma^{\psi_N}(\{\pi_1,\dots,\pi_k\})$ be the $\psi_N$-descent of the indicated cuspidal representations. Let $\pi$ be an irreducible constituent of this descent, and let $s$ be the size of the stabilizer of $\pi$ under $\{1,\theta\}$; in particular, $s=1$ unless $G=\operatorname{SO}(2n)$.

Descent multiplicity conjecture. One has

$$c_\pi^{\psi_N}=\frac{2^{k-1}}{s}.$$

This variant incorporates the possible outer involution for even special orthogonal groups and also covers metaplectic groups. It is a reformulation of the expected Whittaker-coefficient formula and remains open in the stated generality.