Prasanna–Venkatesh rationality conjecture for automorphic cohomology

Let $G$ be a reductive group over $\mathbf{Q}$, let $\pi_f$ be a finite part of a cohomological cuspidal automorphic representation, and let $E(\pi_f)$ be a number field over which the relevant cohomology is defined. Let $\text{\dn{v}}_\pi$ be the motivic cohomology space attached to the adjoint motive, and let $H^*_0(S(G,X),\widetilde V_{\mathbf C})[\pi_f]$ be the corresponding cuspidal cohomology. Prasanna–Venkatesh rationality conjecture. The action of $\bigwedge^{\bullet}\text{\dn{v}}_\pi$ on $H^*_0(S(G,X),\widetilde V_{\mathbf C})[\pi_f]$ preserves the $E(\pi_f)$-rational structure. This predicts that the motivic action is compatible with the rational structures coming from automorphic and topological cohomology.