Uniform finiteness conjecture for Sobolev gaps over the unitary dual

Let $G$ be a real reductive group, let $w$ be a fixed weight on $G$, and let $\mathcal{HC}$ denote the class of Harish-Chandra modules. For a module $V$, write $s(V,w)$ for its Sobolev gap and $\operatorname{Norm}(V,w)$ for the set of corresponding norms; for $\pi\in\widehat G$, write $s(\pi)$ for the associated invariant.

Uniform finiteness conjecture. One has

$$\sup_{\substack{V \in \mathcal{HC}\\ \operatorname{Norm}(V,w)\neq \emptyset}} s(V,w) <\infty.$$

In particular,

$$\sup_{\pi \in \widehat G} s(\pi)<\infty.$$

This is posed as an open problem after finiteness results for important subclasses, including discrete series and minimal principal series. The source gives no resolution of the conjecture.