Quantum sufficiency conjecture for the quantum exceptional category

Let $R$ be a ring and let $\mathsf{QExc}_{R,v,w}$ be the quantum exceptional diagram category. Define the Laurent-polynomial set

$$\mathbf P=\{v,w,\Psi_1,\Psi_2,\Psi_3,\Psi_6,[\lambda],[1-\lambda]\}.$$

Quantum sufficiency conjecture. If every polynomial in $\mathbf P$ is invertible in $R$, then the space of $n$-boundary-point diagrams in $\mathsf{QExc}_{R,v,w}$ is finitely generated as an $R$-module for every $n$. Moreover, the spaces in degrees $0,1,2,3,4$ are generated respectively by the empty diagram, zero, the strand, the trivalent vertex, and the five diagrams specified in the source. This is the quantum analogue of classical sufficiency; the source also conjectures spanning sets through degree $6$.