  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 arXiv:2105.15178 of
the open KPZ stationary measure for all choices of boundary parameters $u,v\in
\mathbb{R}$ (thus going beyond the restriction $u+v\geq 0$ from
arXiv:2103.12253).
