The Martin boundary conjecture for the graph of zigzag diagrams

Let ${\cal Z}$ be the graded graph of zigzag diagrams, with level-${n}$ vertices ${\cal Z}_n$. For a sequence $(\lambda_n\in{\cal Z}_n)$, let $\iota_n(\lambda_n)$ be its image in the compactum ${\cal U}^{(2)}$. For $\mu,\lambda\in{\cal Z}$, let $d(\mu,\lambda)$ be the number of paths from $\mu$ to $\lambda$, let $d(\lambda)$ be the number of paths from the root to $\lambda$, and define

$$K(\mu,\lambda)=\frac{d(\mu,\lambda)}{d(\lambda)}.$$

A sequence is regular when $K(\mu,\lambda_n)$ has a limit for every fixed $\mu$; its limit is denoted by $p(\mu)$. Martin boundary conjecture. (i) A sequence $(\lambda_n\in{\cal Z}_n)$ is regular if and only if the points $\iota_n(\lambda_n)$ converge in the compactum ${\cal U}^{(2)}$. (ii) For $(U_\uparrow,U_\downarrow)\in{\cal U}^{(2)}$, convergence $\iota_n(\lambda_n)\to (U_\uparrow,U_\downarrow)$ is equivalent to $K(\mu,\lambda_n)\to p_{(U_\uparrow,U_\downarrow)}(\mu)$ for all $\mu\in{\cal Z}$. (iii) The Martin boundary of ${\cal Z}$ coincides with its minimal boundary:

$$\partial{\cal Z}={\cal U}^{(2)}.$$

The claim identifies the asymptotic boundary of the zigzag graph with the compactum parametrizing its minimal harmonic functions, and asserts that no additional Martin-boundary points occur. It is presented as a conjectural description of the boundary; the supplied source gives no evidence of a resolution.