Motivic limit conjecture for representation varieties of surface groups

Let $G$ be a connected linear algebraic group over a field $k$, and let $M_g$ be a smooth compact surface of genus $g$. Write $\operatorname{Rep}_G(M_g)$ for the $G$-representation variety of its surface group. Motivic limit conjecture. In the completed Grothendieck ring of stacks,

$$\widehat{\operatorname{K}_0(\operatorname{Stck}_k)},$$

one has

$$\lim_{g\to\infty}\frac{[\operatorname{Rep}_G(M_g)]}{[G^{2g}]}=\frac{[G/[G,G]]}{[G]}.$$

The conjecture is motivated by finite-field heuristics and by the corresponding result established for reductive groups with connected center using the E-polynomial; it remains open for general connected linear algebraic groups.