Gauge-matrix conjecture for monodromy matrices of the extended Lotka–Volterra lattice

Let $N\geq 4$, let $\mathbf{T}_{m;n_1,n_2}(z)$ be the monodromy matrix, and let $L$ be the parity parameter appearing in its construction. Write $\vec{e}_1$ and $\vec{e}_N$ for the first and last standard row vectors, and let

$$\boldsymbol{\mu}_a,\boldsymbol{\mu}_b\in\{\boldsymbol{\mu}_-^{(n_1)},\boldsymbol{\mu}_-^{(n_1-N+1)},\boldsymbol{\mu}_+^{(n_2)},\boldsymbol{\mu}_+^{(n_2-N)},\boldsymbol{\mu}_j\mid j=2,\ldots,m-2\}.$$

Let $\{\mathbf{T}_{m;n_1,n_2}(z)\}_F$ denote the corresponding variety, and let $g$ be the genus of its spectral curve. Gauge-matrix conjecture. There is a gauge matrix $\mathbf{S}$ of the form

$$\mathbf{S}=\begin{cases} \begin{pmatrix} \vec{e}_1\\ \vec{e}_1\boldsymbol{\mu}_a\boldsymbol{\mu}_b^{N-2}\\ \vdots\\ \vec{e}_1\boldsymbol{\mu}_a\boldsymbol{\mu}_b\\ \vec{e}_1\boldsymbol{\mu}_a \end{pmatrix},&\text{if $L$ is even},\\[1em] \begin{pmatrix} \vec{e}_N\boldsymbol{\mu}_a\\ \vec{e}_N\boldsymbol{\mu}_a\boldsymbol{\mu}_b\\ \vdots\\ \vec{e}_N\boldsymbol{\mu}_a\boldsymbol{\mu}_b^{N-2}\\ \vec{e}_N \end{pmatrix},&\text{if $L$ is odd}, \end{cases}$$

such that $\mathbf{S}$ reduces $\{\mathbf{T}_{m;n_1,n_2}(z)\}_F$ to a $g$-dimensional variety $\{\mathbf{M}(z)\}_F$, and the zeros of the associated separation equation are invariant under $\mathbf{S}$. The separation equation has the form specified by the source's separation construction.