Birational affine Weyl group action for the $3\times3$ linear $q$-difference system

Let $A$ satisfy the defining equations referred to as and, with parameters $e_1,\ldots,e_9,\kappa\in\mathbb{C}^\times$ satisfying

$$\kappa\prod_{i=1}^9 e_i=1,$$

and with $e_i\neq e_j$ for $i\neq j$. Quotient by matrix conjugacy, writing $B\simeq C$ when $B$ and $C$ are conjugate. Define $g=(-x/e_1)_\infty(-x/e_2)_\infty(-x/e_3)_\infty$ and choose $\lambda$ such that $q^\lambda=(\kappa e_1e_2e_3)^{-1}$. The transformations $s_i$ act on the parameters by interchanging $e_i$ and $e_{i+1}$ for $i\neq 3$, while $s_3$ interchanges $e_3$ and $e_4$ and acts birationally on the accessory parameters.

Birational affine Weyl group action conjecture. The quotient set of such matrices admits a birational action of the affine Weyl group $W(E_8^{(1)})$, with

$$s_0=g^{-1}\circ mc_\lambda\circ g.$$

The transformations and their incidences are represented by the stated affine Dynkin diagram of type $E_8^{(1)}$.

This claim proposes a birational realization of the affine Weyl group symmetry for the linear $q$-difference system, extending the parameter permutations by the middle-convolution transformation $s_0$. The supplied text does not establish the result or provide evidence resolving the conjectural status of the asserted action.