The non-constant-boundary enumeration conjecture for tree $sl_n$ webs

Let $R_{p,r}(k)$ denote the generalized Catalan number associated with parameters $p,r,k$. For $n \geq 3$, let an $sl_n$ web have a boundary string consisting of one boundary edge labelled $j$ followed by boundary edges labelled $1$; a web lacks an internal cycle when its underlying graph has no cycle away from the boundary. Non-constant-boundary enumeration conjecture. For any $n \geq 3$ and any $1 \leq j \leq n-1$, $(n-2)^k R_{n-1,n-j}(k)$ equals the number of connected $sl_n$ webs that lack an internal cycle and have a boundary string consisting of one $j$ followed by $nk + n - j$ consecutive 1's. This is intended to generalize the preceding proposition for $sl_3$ webs with a non-constant boundary string. If true, it would give a combinatorial interpretation of $R_{p,r}(k)$ for all $k \geq 0$ whenever $1 \leq r \leq p-2$; the supplied text indicates that the result remains unproved.