Uniform fixed-vector growth conjecture for reductive p-adic groups

Let $G$ be a reductive group over a $p$-adic field $F$, let $x$ be a point in its Bruhat–Tits building, and let $K_\ell:=K_{G,x,\ell}$ be the $\ell$th Moy–Prasad subgroup at $x$. Write $q$ for the residue-field cardinality and $d_{GK}(\pi)$ for the Gelfand–Kirillov dimension of a smooth irreducible representation $\pi$ of $G(F)$. Uniform fixed-vector growth conjecture. For every $\epsilon>0$, there is a constant $C=C_{\epsilon,G,F,x}$, independent of $\pi$ and $\ell$, such that

$$\dim(\pi^{K_\ell})\leq C q^{\ell(d_{GK}(\pi)+\epsilon)}.$$

This proposes a uniform upper bound on fixed-vector growth for smooth irreducible representations of general reductive $p$-adic groups, extending the paper's results beyond $\mathrm{GL}_N$. The source gives no resolution status for the conjecture, and the notation $d_{GK}(\pi)$ and the residue cardinality $q$ is only implicit in the supplied context.