The q-lattice representation conjecture for the matrix Fredholm determinant

Let $q$ be the deformation parameter, $t$ the time parameter, and let $\mathcal{D}$ denote the discrete $q$-lattice

$$\mathcal{D}=\mathcal{D}^+_t\sqcup\mathcal{D}^-_1=\{tq^n\mid n\in\mathbb{Z}_{>0}\}\sqcup\{q^n\mid n\in\mathbb{Z}_{<0}\}.$$

Let $\mathcal{D}$ be the matrix Fredholm determinant defined earlier in the paper. q-lattice determinant conjecture. The determinant can be written as a determinant on the $q$-lattice $\mathcal{D}$. This would identify the analytic Fredholm determinant with a discrete determinant supported on the indicated lattice; the supplied text gives no proof or resolution.