The generalized theta-to-q difference-equation conjecture

Let $m$-periodic variables $\theta_i^t$ satisfy the generalized discrete Toda equations referred to as, and define $q_{j,i}^t$ from them via. Theta-to-q conjecture. The quantities $q_{j,i}^t$ satisfy all the difference equations. This conjecture extends Proposition; it asserts that the theta-function parametrization produces solutions of the full system of difference equations, beyond the previously established case.