The MERW transition conjecture for the growing pyramidal model

Let $(x,y,z)$ be a permissible pyramidal configuration, and let $(u,v,w)$ be obtained from it by incrementing one coordinate by $1$. Associate to a configuration $(x,y,z)$ the differences $(y-x,y-z)$. Let $H$ be the unique non-negative function on $\mathbb{Z}^2$ such that

$$H_{i,j}=0\quad\text{for }i,j<0,$$

$H_{0,0}=1$, and, for all $i\geq 0$ or $j\geq 0$,

$$3H_{i,j}=H_{i-1,j}+H_{i,j-1}+H_{i,j}.$$

MERW transition conjecture. The unique maximal entropy random walk for the growing pyramidal model depicted by the Kreweras-walk correspondence has transition probabilities

$$p((x,y,z);(u,v,w))=\frac{H_{v-u,v-w}}{3H_{y-x,y-z}}.$$

This conjecture identifies the maximal entropy random walk through the unique non-negative harmonic function for the associated Kreweras random walk in the three-quarter plane. The surrounding discussion indicates that the required asymptotics, and hence this characterization, are motivated by known harmonic-function results but does not establish the conjectured transition rule.