The affine Laumon partition function conjecture for the non-stationary difference equation

Let $N$ be a positive integer, let $x_i$, $b_i$, $d_i$, and $\overline{d}_i$ be parameters, and let $\mathcal{H}^{{\widehat{\mathfrak{gl}}_N}}(x_i;b_i,d_i,\overline{d}_i,q,\kappa)$ be the ${\widehat{\mathfrak{gl}}_N}$ non-stationary difference operator. Define

$$\psi=\sum_{\theta_1,\ldots,\theta_N=0}^{\infty}c_{\theta_1,\ldots,\theta_N}x_1^{\theta_1}\cdots x_N^{\theta_N},\qquad c_{0,\ldots,0}=1.$$

The affine Laumon partition function provides a solution of the non-stationary difference equation

$$\mathcal{H}^{{\widehat{\mathfrak{gl}}_N}}(x_i;b_i,d_i,\overline{d}_i,q,\kappa)\psi=\psi,$$

where $\psi$ is the ${\widehat{\mathfrak{gl}}_N}$ Laumon partition function with the parametrization

$$\psi=\mathcal{Z}_{\mathrm{AL}}^{{\widehat{\mathfrak{gl}}_N}}\left(\left.\left.\begin{array}{ccc} \frac{q\kappa b_N}{d_N},\frac{q\kappa b_1}{d_1},\ldots,\frac{q\kappa b_{N-1}}{d_{N-1}}\\b_1,b_2,\ldots,b_N\\frac{b_1}{\overline{d}_1},\frac{b_2}{\overline{d}_2},\ldots,\frac{b_N}{\overline{d}_N}\end{array}\right|\sqrt{\frac{b_2d_1\overline{d}_1}{q\kappa b_1}}x_1,\sqrt{\frac{b_3d_2\overline{d}_2}{q\kappa b_2}}x_2,\ldots,\sqrt{\frac{b_1d_N\overline{d}_N}{q\kappa b_N}}x_N\right|q,\kappa\right).$$

This conjecture extends the established $A_1^{(1)}$ case to type $A_{N-1}^{(1)}$ and predicts a connection between affine Laumon partition functions, non-stationary difference equations, and quantum affine algebraic structures. The general statement is presented as the paper's main claim, while the supplied text gives no resolution beyond the previously proved $A_1^{(1)}$ case.