Linear dependence of the Sobolev gap on the weight parameter

Let $G$ be a real reductive group, let $w$ be a weight on $G$, and let $V$ be a Harish-Chandra module such that $\operatorname{Norm}(V,w)\neq\emptyset$. For $t\geq 0$, let $w_t$ be the modified weight defined in the surrounding text.

Weight-dependence conjecture. There exists a constant $C=C_G>0$ depending only on $G$ such that

$$s(V,w_t) \leq s(V,w) + C t \qquad (t\geq 0).$$

This conjecture asks for a uniform linear bound on how the Sobolev gap changes under the specified modification of the weight. The supplied source does not give evidence of resolution.