The unified even-modulus eta-function identity

Let $k,N$ be positive integers and set $n=\lfloor N/2\rfloor$. For $1\leq a,b\leq N-1$, let $C_{ab}$ be the Cartan integers of $\operatorname{A}_{N-1}$, and define $M_i^{(a)}=m_i^{(a)}+\cdots+m_{k-1}^{(a)}$. Let $\boldsymbol{\rho}^{\star}=(0,1,\dots,n-1)$ and let $\boldsymbol{\rho}$ denote the vector used in the odd-$N$ identity. The sums are over $\boldsymbol{v}\in\mathbb Z^n$ satisfying the stated congruence conditions, and $(q)_r$, $(q^2;q^2)_r$, $\xi$, and $\chi$ have the meanings defined in the source. Unified even-modulus eta-function conjecture. The two expressions in the source are equal:

$$\sum \frac{q^{\frac{1}{2}\sum_{a,b=1}^{N-1}\sum_{i=1}^{k-1} C_{ab}M_i^{(a)}M_i^{(b)}}}{\prod_{a=1}^{N-1}\bigl(\prod_{i=1}^{k-2}(q)_{m_i^{(a)}}\bigr)(q^2;q^2)_{m_{k-1}^{(a)}}} =\frac{1}{(q)_{\infty}^{N(N-1)/2}}\sum \xi(\boldsymbol{v}/\boldsymbol{\rho}^{\star})(-1)^{\tfrac{\lvert\boldsymbol{v}\rvert-\lvert\boldsymbol{\rho}^{\star}\rvert}{2k+N-2}}q^{\tfrac{||\boldsymbol{v}||^2-||\boldsymbol{\rho}^{\star}||^2}{2(2k+N-2)}}$$

for even $N$, with $v_i\equiv\rho_i^{\star}\pmod{2k+N-2}$, and

$$\frac{1}{(q)_{\infty}^{N(N-1)/2}}\sum \chi(\boldsymbol{v}/\boldsymbol{\rho})q^{\tfrac{||\boldsymbol{v}||^2-||\boldsymbol{\rho}||^2}{2(2k+N-2)}}$$

for odd $N$, with $v_i\equiv\rho_i\pmod{2k+N-2}$. This conjecture unifies Bressoud's identity for $p=k$ with Macdonald's eta-function identities; the source gives no general proof, so the claim remains open.