Half-space Beta RWRE Gamma-distribution conjecture

Fix $t\in\mathbb Z$. In the half-space Beta random walk in random environment, define

$$\mathsf Z_t(x)=\lim_{L\to\infty}\sum_{z=\max\lbrace 0,x-L\rbrace}^{x+L}\mathsf P_{t-L,t}(z,x),$$

for $x\in\mathbb Z_{\geqslant 0}$. Half-space Gamma-distribution conjecture. The limit exists almost surely for every $x\in\mathbb Z_{\geqslant 0}$, and the variables $\left(\mathsf Z_t(x)\right)_x$ are independent with distribution

$$\mathsf Z_t(x)\sim \begin{cases} \frac{1}{2\mu}\operatorname{Gamma}(2\mu) & \text{if }x\geqslant 2,\\\\ \frac{1}{2\mu}\operatorname{Gamma}(\mu+\eta) & \text{if }x=1,\\\\ \frac{1}{2\mu}\operatorname{Gamma}(\eta) & \text{if }x=0. \end{cases}$$

Unlike the full-space setting, the half-space environment is neither iid nor spatially translation invariant, and trapping at the boundary can substantially affect return probabilities. The conjecture proposes that these effects nevertheless lead to the displayed independent, explicitly parameterized Gamma laws.