Finite-dimensionality conjecture for the module generated by closed surfaces

Let $G$ be a complex algebraic group, and let $\mathrm{K}\operatorname{M}_{G}$ denote the ambient module of motivic data associated with $G$. Define

$$W_G=\left\langle \mathfrak{Z}(L)^g(i_!\mathbb{Q}_0)\right\rangle_{g=0}^{\infty}\subseteq \mathrm{K}\operatorname{M}_{G},$$

the submodule generated by the image of the unit under all closed surfaces with connected boundary. Finite-dimensionality conjecture. The module $W_G\subseteq \mathrm{K}\operatorname{M}_{G}$ is finitely dimensional.

The conjecture would reduce the computations of the associated topological quantum field theory to finitely many generators, despite the possibility that the full module $\mathrm{K}\operatorname{M}_{G}$ is infinitely generated. It is motivated by known finite-generation computations for $G=\operatorname{SL}(2,\mathbb{C})$ and $G=\operatorname{PGL}(2,\mathbb{C})$, but the stated result is presented as an expectation for a general class of algebraic groups in the non-parabolic case.