Weil's product formula for the moduli stack of $G$-bundles

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ be a smooth affine group scheme with semisimple simply connected generic fiber and connected fibers, and let $\mathrm{Bundle}_G(X)$ denote the moduli stack of $G$-bundles. For $x\in X$, let $\mathrm{Bundle}_G(\{x\})$ be the corresponding local classifying stack and let $\kappa(x)$ be the residue field. Weil's conjecture. The following equality is well-defined:

$$\frac{|\mathrm{Bundle}_G(X)(\mathbb{F}_p)|}{q^{\mathrm{dim}\mathrm{Bundle}_G(X)}}=\prod_{x\in X}\frac{|\mathrm{Bundle}_G(\{x\})(\kappa(x))|}{q^{\mathrm{dim}\mathrm{Bundle}_G(\kappa(x))}}.$$

The claim is presented as the function-field analogue of the Tamagawa-number product formula, but the supplied text does not state whether it is proved.