Super Catalan number recurrence from generalized action graph path counts

Let $G_n$ be the action graph associated with $S(0,n)$, and let $K_{\ell,v,n}$ be the number of paths of length $\ell$ in $G_n$ that start at a vertex labeled $v$ and end at a vertex labeled $n$. Super Catalan computation conjecture. The subsequent super Catalan number can be computed from the $n$-table of its previous action graph via

$$S(0,n+1) = \displaystyle \sum_{\ell=0}^{n} \left(\frac{2}{2^\ell} \sum_{v=0}^{n} K_{\ell,v,n}\right).$$

If the path-count recurrence conjecture holds, the authors believe this formula computes $S(0,n+1)$ from the $n$-table and establishes that $G_n$ satisfies Axiom1. The formula is presented as part of an approach whose general validity remains open.