Spherical matrix-coefficient lower-bound conjecture

Let $G$ be the ambient group, $K$ a maximal compact subgroup, and $\Pi(G)_{\operatorname{sph}}$ the set of unitary irreducible spherical representations of $G$. For $g\in G$, let $S(g)$ be the $K$-bi-invariant function in the Cartan integration formula, and let $l(g)$ be the associated length. For $(\pi,V)\in\Pi(G)_{\operatorname{sph}}$ with $p(\pi)>2$, choose a $K$-fixed vector $v\in V$ with $\|v\|=1$. Spherical matrix-coefficient conjecture. There exist $D>0$ and $L>0$ such that, for every $\epsilon>0$ and every such $(\pi,V)$, if $d_0>D$, then in the non-Archimedean case

$$\int_{l(g)\le d_0}S(g)\left|\left\langle v,\pi(g)v\right\rangle\right|^2\,dg\gg_\epsilon q^{2d_0(1-1/p(\pi)-\epsilon)},$$

and in the Archimedean case

$$\int_{l(g)\le d_0}S(g)\left|\left\langle v,\pi(g)v\right\rangle\right|^2\,dg\gg_\epsilon(\lambda(\pi)+1)^{-L}q^{2d_0(1-1/p(\pi)-\epsilon)}.$$

Such a lower bound would provide the desired optimal spherical test functions for the density framework. The source compares it with partial results for $G=\operatorname{PGL}_n$ and states the assertion as a conjecture; no general resolution is given.