Conjecture on convergence of integer-valued Lipschitz functions under general boundary conditions

Let $\mu_{n,d}^S$ denote the Gibbs measure for integer-valued Lipschitz functions on the depth-$n$ $d$-ary tree with boundary condition given by a finite set $S\subset\mathbb{Z}$.

Convergence conjecture.

$$\lim_{n \to \infty}\mu_{n,2}^S$$

exists for all finite $S$. Also,

$$\lim_{n \to \infty}\mu_{2n,d}^S$$

exists for all finite $S$ and all $d\ge 2$.

The conjecture proposes convergence for every finite boundary condition when $d=2$, and along even depths for every branching parameter $d\ge2$. The paper establishes convergence in several regimes, including the cases considered for small $d$, while for large $d$ it obtains convergence along even sequences; the full assertion remains open.