Central measures of the jump graph for Young--Fibonacci graph

For fixed $k$, we consider the subgraph $Y{F}_k=(V_k,E_k)$ of the famous Young–Fibonacci graph formed by the words with at most $k$ 2-s. The jump graph is a graded graph is defined as follows: each level is identified with $V_k$, and an edge between two vertices $(v_1,i)$ and $(v_2,i+1)$, $v_1,v_2\in V_k$, of neighbouring levels us drawn iff $v_2$ is a descendant of $v_1$ in $Y{F}_k$. The goal of this paper is to describe all central measures on the path space of $Y{F}_k$.