Gnedin–Olshanski conjecture on the Martin boundary of the zigzag graph

Let $\mathcal{Z}$ be the zigzag graph, let $\partial_{M}\mathcal{Z}$ and $\partial_{\min}\mathcal{Z}$ denote its Martin and minimal entrance boundaries, and let $\mathcal{U}^{(2)}$ be the space of pairs $(U_{\uparrow},U_{\downarrow})$ of disjoint open subsets of $[0,1]$ arising in the boundary parametrization. For each composition $\lambda$ of $n$, define $U_{\lambda}=(U_{\uparrow}(\lambda),U_{\downarrow}(\lambda))\in\mathcal{U}^{(2)}$ by

$$U_{\uparrow}(\lambda)=\operatorname{int}\left(\bigcup_{i\notin\operatorname{des}(\lambda)} I_i\right),\qquad U_{\downarrow}(\lambda)=\operatorname{int}\left(\bigcup_{i\in\operatorname{des}(\lambda)} I_i\right),$$

where $I_s=[(s-1)/(n-1),s/(n-1)]$ for $s\leq n-1$. Write $K_{\mu}(\lambda)$ for the Martin kernel indexed by $\mu\in\mathcal{Z}$, and $p_{(U_{\uparrow},U_{\downarrow})}(\mu)$ for the corresponding boundary transition probability. Gnedin–Olshanski conjecture. For every sequence $(\lambda_n)_{n\geq1}$, the following hold: (a) $(\lambda_n)$ belongs to $\partial_M\mathcal{Z}$ if and only if $U_{\lambda_n}$ converges in $\mathcal{U}^{(2)}$; (b) $U_{\lambda_n}\to(U_{\uparrow},U_{\downarrow})$ in $\mathcal{U}^{(2)}$ if and only if $K_{\mu}(\lambda_n)\to p_{(U_{\uparrow},U_{\downarrow})}(\mu)$ for every $\mu\in\mathcal{Z}$; and (c) the Martin boundary coincides with the minimal boundary, $\partial_M\mathcal{Z}=\mathcal{U}^{(2)}$. The conjecture identifies all Martin-boundary limits with the already parametrized minimal boundary and characterizes convergence through both the geometric parameters $U_{\lambda_n}$ and the Martin kernels. The source attributes the problem to Conjecture 45 of Gnedin and Olshanski; no resolution is given here.