  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 top marginal of a two-layer Markov chain
with an explicit description, which can be interpreted as two random walks
conditioned softly never to intersect. This limiting law is a stationary
measure for the polymer on the anti-diagonal path.
  The starting point of our analysis is an embedding of the free energy into
the half-space log-gamma line ensemble recently constructed by Barraquand,
Corwin, and Das. Given the Gibbsian line ensemble structure, the main
contribution of our work lies in developing a route to access and prove
convergence to stationary measures via line ensemble techniques. Our argument
relies 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.
