The rational cohomology conjecture for moduli stacks of principal bundles

Let $X$ be a connected smooth complex projective variety of complex dimension $n$, and let $G$ be a semisimple complex algebraic group. Write $d_j=\mathrm{dim\,} H^j(X,\mathbb Q)$ for $0\leq j\leq 2n$, let $V$ denote the graded rational cohomology algebra associated with the classifying space of $G$, and let $\mathscr{B}un_{G,X}^0$ be any connected component of the moduli stack $\mathscr{B}un_{G,X}$.

Rational cohomology conjecture. The rational cohomology of $\mathscr{B}un_{G,X}^0$ is

$$H^*(\mathscr{B}un_{G,X}^0,\mathbb Q) \cong \left(\bigotimes_{j=0}^n (\operatorname{Sym} V[2j])^{\otimes d_{2j}}\right)\otimes\left(\bigotimes_{j=1}^n (\Lambda V[2j-1])^{\otimes d_{2j-1}}\right),$$

and this isomorphism is induced by the pullback map from the classifying space. In particular, it is a direct sum of Hodge-Tate structures.

This generalizes the preceding result for varieties whose cellular decompositions have no odd-dimensional cells. The claim predicts that the rational cohomology depends on the cohomology dimensions of $X$ through the symmetric and exterior contributions, but its resolution is not established in the supplied text.