The inert quadratic order conjecture for finitized Coh zeta functions

Let $q$ be a prime power, let $m\geq 1$, and let

$$R'_{2,2m}:={\mathbb F}_q[[T]]+T^m{\mathbb F}_{q^2}[[T]]$$

be the inert quadratic order in ${\mathbb F}_{q^2}((T))/{\mathbb F}_q((T))$. For an $R$-module $M$, write

$$\zeta_M^R(s):=\sum_{L\subseteq_R M}(M:L)^{-s},$$

where the sum is over finite-index submodules, and define the finitized Coh zeta function by $\widehat{\zeta}_{R,n}(s):=\zeta_{R^n}^R(s+n)$. Set $t=q^{-s}$, and let $\mathbf{Br}_n(q,t;2m+2)$ denote the $t$-deformed Bressoud polynomial

$$\mathbf{Br}_n(q,t;2m+2):=(q;q)_n\sum_{n_1,\dots,n_m}\frac{q^{\sum n_i^2}t^{2\sum n_i}}{(q;q)_{n-n_1}(q;q)_{n_1-n_2}\cdots(q;q)_{n_m}(-tq;q)_{n_m}}.$$

The inert quadratic order conjecture. For each $m\geq 1$,

$$\widehat{\zeta}_{R'_{2,2m},n}(s)=\frac{1}{(tq^{-1};q^{-1})_n}\mathbf{Br}_n(q^{-1},t;2m+2).$$

This conjecture supplies the missing inert case in the unified description of finitized Coh zeta functions for quadratic orders and identifies the direct $t$-deformation with a Bressoud sum; its resolution would also clarify the structural symmetry observed in the ramified case.