Alev's uniqueness conjecture for simple graphs solving the B_n equation

Let $B_n$ be the type-$B_n$ case, and let $S^W_{\mathfrak{sl}_2}$ be represented by graphs, with polynomials of degree $4(n-1)$ corresponding to linear combinations of graphs with $n-1$ edges. Consider the equation referred to in the source as Equation. A simple graph is the graph formed from the linear graph without loops or isolated vertices, with $n-1$ edges and $n$ vertices, together with corrective graphs having the same number of edges and non-isolated vertices.

Simple-graph uniqueness conjecture. For every integer $n\geq 2$, there exists a unique polynomial of degree $4(n-1)$ in $S^W_{\mathfrak{sl}_2}$ solving the equation. It is represented by the $n$th simple graph.

This conjecture concerns the proposed canonical solution at the top relevant degree and its graph-theoretic form. The source gives the construction and terminology but does not state a general proof.