The spanning conjecture for the sets of bounded-genus surfaces

Let $A(n)$ be the space defined by the two-dimensional topological theory, and let $\mathscr{A}^m(n)$ be the set of surface diagrams whose connected components satisfy the genus bound encoded by $m$. Spanning conjecture. The set $\mathscr{A}^m(n)$ spans $A(n)$.

The claim would identify these bounded-genus surface diagrams as a spanning family for $A(n)$. The supplied context proves linear independence for generic parameters but does not establish the spanning assertion or give evidence that it has been resolved.