Classical sufficiency conjecture for the exceptional diagram category

Let $R$ be a ring and let $\boldsymbol{\lambda}$ be a parameter such that the category $\mathsf{Exc}_{R,\lambda}$ is defined; write $\mathsf{Exc}_{R,\lambda}(n)$ for its space of diagrams with $n$ boundary points. Classical sufficiency conjecture. If a certain finite set of nonzero polynomials in $\lambda$ is invertible in $R$, then $\mathsf{Exc}_{R,\lambda}(n)$ is finitely generated as an $R$-module for every $n$. Moreover, $\mathsf{Exc}_{R,\lambda}(0)$ is spanned by the empty diagram, $\mathsf{Exc}_{R,\lambda}(1)$ is the zero module, $\mathsf{Exc}_{R,\lambda}(2)$ is spanned by the strand, $\mathsf{Exc}_{R,\lambda}(3)$ is spanned by the trivalent vertex, and $\mathsf{Exc}_{R,\lambda}(4)$ is spanned by the five diagrams specified in the source. The conjecture would give finite diagrammatic control of the classical exceptional family; the source also notes that specific spanning sets are conjectured through $n=6$, while the required finite set of polynomials is not explicitly identified in the statement.