Uniform compactness conjecture for normalized matrix coefficients on real spherical spaces

Let $Z=G/H$ be a real spherical space with compression cone $A_Z^-$. Let $V$ be a Harish-Chandra module for $(\mathfrak{g},K)$, let $V^\infty$ be its moderate-growth smooth globalization, and let $(V^{-\infty})^H$ denote the space of $H$-invariant distribution vectors. For a fixed $K$-type $\tau$, write $V[\tau]$ for its $\tau$-isotypic component, let $\Lambda_V$ be the $H$-spherical exponent of $V$, and let $d_V\in\mathbb{N}$ be its logarithmic exponent. For $v\in V$ and $\eta\in(V^{-\infty})^H$, write $m_{v,\eta}$ for the corresponding generalized matrix coefficient. Uniform compactness conjecture. For every $K$-type $\tau$, constant $C>0$, and compact subset $\Omega\subset G$, there exists a compact set $\Omega_A\subset A_Z^-$ such that, for every Harish-Chandra module $V$ with $\|\Lambda_V\|\leq C$, every $v\in V[\tau]$, and every $\eta\in(V^{-\infty})^H$,

$$\max_{a\in A_Z^-,\,g\in\Omega}|m_{v,\eta}(ga\cdot z_0)|a^{-\Lambda_V}(1+\|\log a\|)^{-d_V} = \max_{a\in\Omega_A,\,g\in\Omega}|m_{v,\eta}(ga\cdot z_0)|a^{-\Lambda_V}(1+\|\log a\|)^{-d_V}.$$

This conjecture would imply Hypothesis A and give uniform control of the normalized growth of matrix coefficients across families of Harish-Chandra modules with bounded spherical exponents. The source presents it as an open direction in harmonic analysis on real spherical spaces, with applications to lattice counting; no resolution is supplied here.