Uniformity conjecture for local representation zeta functions of type A groups

Let $n\in\mathbb{N}_{\geq 2}$. For a divisor $\iota(v)=\gcd(q_v-\varepsilon(v),n)$ of $n$, where $\varepsilon(v)\in\{1,-1\}$ records whether the relevant quadratic extension splits at $v$, let $k$ be a number field and let $\mathbf{H}$ be a connected, simply-connected absolutely almost simple $k$-algebraic group of type~$\mathsf{A}_{n-1}$. If $\mathbf{H}$ is an outer form, let $K$ be the quadratic extension appearing in its definition; if it is an inner form, put $K=k$. The notation $q_v$ denotes the residue cardinality of ${\mathcal{O}}_v$. Uniformity conjecture. There exist finite index sets $I$ and $J$, polynomials $f_{\iota,\varepsilon,i},g_{\iota,\varepsilon,i}\in\mathbb{Q}[t]$ for $(\iota,\varepsilon,i)\in\operatorname{Div}(n)\times\{1,-1\}\times I$, and non-negative integers $A_j,B_j$ for $j\in J$, together with a finite set $S$ of places of $k$, containing all archimedean places and depending on $\mathbf{H}$, such that for every place $v\notin S$,

$$\zeta_{\mathbf{H}({\mathcal{O}}_v)}(s)=\frac{\sum_{i\in I}f_{\iota(v),\varepsilon(v),i}(q_v)\,g_{\iota(v),\varepsilon(v),i}(q_v)^{-s}}{\prod_{j\in J}(1-q_v^{A_j-B_js})}.$$

This conjecture generalizes the established type $\mathsf{A}_2$ case and predicts a uniform rational form for almost all local factors, with dependence only on the residue cardinality and the indicated arithmetic data; its general case remains open.