Stationary-measure and phase-diagram conjecture for the half-space log-Gamma polymer

Let $Z(x,n)$ be the half-space log-Gamma polymer partition function governed by the recurrence in the source, and let $\mathcal{T}_{\beta,t}$ denote the stationary spatial process with parameters $\beta$ and $t$. For given $\alpha \in \mathbb{R}_{>0}$ and $\beta,t \in \mathbb{R}$ with $\alpha>t>0$ and $\beta>-t$, the family $\\{\mathcal{T}_{\beta,t}\\}$ consists of all extremal stationary measures. If the initial data satisfy $\lim_{x\to\infty}\log Z(x,1)/x=-d\in\mathbb{R}$ and $t=\alpha-\psi^{-1}(d)$, then the normalized process has the following weak limits as $N\to\infty$: in the high-density regime, $\beta\geq-t$ and $t\geq0$, it converges to $\mathcal{T}_{\beta,t}$; in the maximal-current regime, $\beta\geq0$ and $t\leq0$, it converges to $\mathcal{T}_{\beta,0}$; and in the low-density regime, $-t\geq\beta$ and $\beta\leq0$, it converges to $\mathcal{T}_{-t,t}$, a $\operatorname{Gamma}^{-1}(\alpha+t)$ multiplicative random walk.

Stationary-measure and phase-diagram conjecture. The stated family gives all extremal stationary measures, and the three parameter regimes give the corresponding weak limits of $\left(Z(x+N,N)/Z(N,N)\right)_{x\in\mathbb{Z}_{\geq0}}$. This identifies the limiting spatial process for each phase of the half-space model. The source presents this as a conjectural phase diagram and does not provide evidence of resolution.