Stationary measures for integrable polymers on a strip

We prove that the stationary measures for the free-energy increment process for the geometric last passage percolation (LPP) and log-gamma polymer model on a diagonal strip is given by a marginal of a two-layer Gibbs measure with a simple and explicit description. This result is shown subject to certain restrictions on the parameters controlling the weights on the boundary of the strip. However, from this description and an analytic continuation argument we are able to access the stationary measure for all boundary parameters. Taking an intermediate disorder limit of the log-gamma polymer stationary measure in a strip we readily recover (modulo convergence of the polymer to the open KPZ equation, Conjecture 4.2) the conjectural description from (Barraquand, Le Doussal in Europhys. Lett. 137(6):61003, 2022) of the open KPZ stationary measure for all choices of boundary parameters $u,v\in \mathbb{R}$ $u,v\in \mathbb{R}$ (thus going beyond the restriction $u+v\ge 0$ $u+v\geq 0$ from (Corwin, Knizel in Stationary measure for the open KPZ equation, 2021, arXiv:2103.12253)).