Metaplectic-lift conjecture for the Plancherel decomposition of invertible symmetric matrices

Let $G$ be the group acting on the set $X$ of invertible symmetric matrices by $x\cdot g=\,{}^tgxg$, and write the Plancherel decomposition as

$$L^2(X)=\int_{\operatorname{Temp}(G)}\pi^{\oplus m(\pi)}d\pi.$$

Let $G'$ be a two-fold Kazhdan–Patterson covering group of $G$. For a genuine irreducible tempered representation $\pi'$ of $G'$ and an irreducible tempered representation $\pi$ of $G$, call $\pi$ the metaplectic lift of $\pi'$ when their Harish–Chandra characters satisfy the relevant trace formula. For fixed $\pi$, let $(\iota_{G'}^{G})^{-1}(\pi)$ be the finite set of genuine irreducible tempered representations of $G'$ lifting to $\pi$, and let $d_{\operatorname{Wh}}(\pi')$ denote the Whittaker dimension of $\pi'$. Metaplectic-lift conjecture. Under these settings, (1) $\pi\in\operatorname{Temp}(G)$ occurs in the Plancherel decomposition if and only if $\pi$ is the metaplectic lift for some $\pi'$, and (2)

$$m(\pi)=\left|(\iota_{G'}^{G})^{-1}(\pi)\right|\cdot d_{\operatorname{Wh}}(\pi')^2,$$

where $\pi'$ is any genuine irreducible tempered representation of $G'$ lifting to $\pi$. This conjecture predicts both the tempered spectrum of the symmetric space and its multiplicities in terms of metaplectic correspondence and Whittaker dimensions; its resolution is not specified in the source.