The motivic Tamagawa conjecture for simply connected groups

Let $C$ be a smooth projective geometrically connected curve over $k$, of genus $g$, and let $G$ be a split semisimple connected algebraic group over $k$. Denote by ${\mathfrak{Bun}}^{}_{G,C}$ the moduli stack of $G$-torsors on $C$, by $\mu({\mathfrak{Bun}}^{}_{G,C})$ its motive, and by $\widehat{K}_0^G(\operatorname{Var}_k)$ the relevant completed equivariant Grothendieck ring. Let $d_1,\ldots,d_r$ be the numbers one higher than the exponents of $G$, and let $Z(C,u)$ be the motivic zeta function of $C$. The motivic Tamagawa conjecture. If $G$ is simply connected, then

$$\mu({\mathfrak{Bun}}^{}_{G,C})={\mathbb L}^{(g-1)\dim G}\prod_{i=1}^r Z(C,{\mathbb L}^{-d_i})$$

in $\widehat{K}_0^G(\operatorname{Var}_k)$. This is proposed as a motivic version of Weil's Tamagawa number conjecture. Its counting-measure specialization over a finite field is equivalent to the Tamagawa-number formula, and the paper gives evidence from Poincare characteristics, Harder's and Ono's results, and the case of $C={\mathbb P}^1$; the general motivic identity remains conjectural.