The planar bipartite graph basis conjecture for $S^{(k,k,k,1^{n-3k})}$

Let $A$ be the set of planar bipartite graphs embedded in a disk satisfying the following conditions: there are $n$ boundary vertices, all in the same part of a bipartition; every interior vertex is connected to a boundary vertex; every interior vertex in the same bipartition part as the boundary vertices has degree $3$ and is called negative; every other interior vertex has degree at least $3$ and is called positive; the number of positive interior vertices minus the number of negative interior vertices is exactly $k$; and there are no cycles of length less than $6$. Let $S^{(k,k,k,1^{n-3k})}$ denote the corresponding Specht module. The planar bipartite graph basis conjecture. The cardinality of $A$ equals the dimension of $S^{(k,k,k,1^{n-3k})}$:

$$|A|=\dim S^{(k,k,k,1^{n-3k})}.$$

This proposes a combinatorial model, analogous to the $A_2$-web basis for $S^{(k,k,k)}$, for a basis of the indicated Specht module. The statement is presented as a potential candidate rather than an established result, and its resolution is not given in the source.