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

Let $R_{p,r}(k)$ denote the generalized Catalan number associated with parameters $p,r,k$. An $sl_n$ web is a diagrammatic object for the representation theory of $U_q(sl_n)$; a web lacks an internal cycle when its underlying graph has no cycle away from the boundary, and its boundary string records the labels on its boundary edges. Constant-boundary enumeration conjecture. For any $n \geq 3$, $(n-2)^k R_{n+1,n-1}(k)$ equals the number of connected $sl_n$ webs that lack an internal cycle and have a boundary string with $n(k+1)$ total 1's. This conjecture generalizes the preceding enumeration result for $A_2$ (or $sl_3$) webs to tree webs for $sl_n$. It remains open because the coral-diagram interpretation requires proving the number of valid, non-equivalent choices for the base and each $(n+1)$-star, as well as showing that distinct pieces do not interact to produce further relations.