Path-count recurrence for generalized action graphs of the super Catalan numbers

Let $G_n$ be the graph associated with $S(0,n)$, and let $K_{\ell,v,n}$ denote the number of paths of length $\ell$ in $G_n$ that start at a vertex labeled $v$ and end at a vertex labeled $n$. Path-count recurrence conjecture. The number of paths of length $\ell$ in $G_{n+1}$ that start at a vertex labeled $v$ and end at a vertex labeled $n+1$ is given by

$$K_{\ell,v,n+1} = \displaystyle \sum_{i=0}^{n+1-\ell-v} \frac{2}{2^i} K_{\ell-1+i,v,n}.$$

The recurrence is intended to compute the entries of the $n$-tables without explicitly counting every path. It has been checked through the $7$-tables, but the general case remains unproved.