Hidden diagonal integrability of q-Hahn vertex model and Beta polymer model

We study a new integrable probabilistic system, defined in terms of stochastic colored vertex model on square lattice. The main distinctive feature our is family parameters attached to diagonals rather than rows or columns, like other similar models. Because these the previously known results about models cannot be directly applied, but nevertheless integrability remains, and we prove explicit integral expressions for $q$-deformed moments (colored) height functions model. Following techniques can interpreted as $q$-discretization Beta polymer from arXiv:1503.04117 with parameters, also diagonals. To demonstrate how respect diagonal works, extend Tracy-Widom large-scale fluctuations