Convergence to stationary measures for the half-space log-gamma polymer

We consider the point-to-point half-space log-gamma polymer model in the unbound phase. We prove that the free energy increment process on the anti-diagonal path converges to the marginal of a two-layer Markov chain with an explicit description. The limiting law is a stationary measure for the polymer on the anti-diagonal path. The existence and identification of the limit rely on a description of the limiting behavior of two softly non-intersecting random walk bridges around their starting point, a result established in this paper that may be of independent interest.