Lattice comparison conjecture for representation growth

Let

$$H=\prod_{i=1}^{\ell}G_i(K_i),$$

where each $K_i$ is a local field, each $G_i$ is an absolutely almost simple $K_i$-group, and

$$\sum_{i=1}^{\ell}\operatorname{rk}_{K_i}(G_i)\geq 2.$$

Let $\rho(\Gamma)$ denote the abscissa of convergence of the representation zeta function of a lattice $\Gamma$. Lattice comparison conjecture. For any two irreducible lattices $\Gamma_1$ and $\Gamma_2$ in $H$,

$$\rho(\Gamma_1)=\rho(\Gamma_2).$$

The conjecture is motivated by examples for powers of $\operatorname{SL}_2$ and predicts that representation growth is determined by the ambient higher-rank semisimple group rather than by the particular irreducible lattice; the source presents supporting results but leaves the general assertion open.