Conjecture on extremal stationary measures for the half-space log-gamma polymer

Let $\alpha>0$, $u>-\alpha$, and let $z_{u,v}$ be the explicitly constructed stationary processes indexed by $v\in(-\alpha,\min(u,0)]$ for the half-space log-gamma polymer. For initial data $z(k,0)$ satisfying $\lim_{k\to\infty}\log z(k,0)/k=-d$, define $v=\psi^{-1}(d)-\alpha$, where $\psi^{-1}(d)$ is the unique positive root of $\psi(z)=d$. Extremal stationary-measure conjecture. The family $\{z_{u,v}\}_{v\in(-\alpha,\min(u,0)]}$ constitutes all extremal stationary measures, and the normalized process $z(\cdot,m)/z(m,m)$ converges weakly to the phase-dependent limit: $z_{u,v}$ or $z_{u,0}$ when $u\geqslant0$, and $z_{u,v}$ or $z_{u,u}$ when $u\leqslant0$, with the alternatives determined respectively by $v\leqslant0$ versus $v\geqslant0$, and $v\leqslant u$ versus $v\geqslant u$. This is presented as the discrete counterpart of the KPZ stationary-measure conjecture.