The rationality conjecture for the archimedean motivic action

Let $L$ be a finite Galois extension through which the Artin representation associated to $f$ factors, let $E$ be a coefficient field, and choose embeddings $\tau:L\rightarrow\mathbb{C}$ and $\iota:E\rightarrow\mathbb{C}$. Let $U_f$ be the Stark unit group and let the partial complex-conjugation action identify $U_f^\vee\otimes_\iota\mathbb{C}$ with the corresponding archimedean cohomological operators.

Archimedean rationality conjecture. The action of $U_f^\vee\otimes E\subseteq U_f^\vee\otimes_\iota\mathbb{C}$ on $H^\ast(X_\mathbb{C},\mathcal{E}_{\underline 1,1})_f$ preserves the rational structure $H^\ast(X,\mathcal{E}_{\underline 1,1})_f\otimes_{E_f}E$.

This is the archimedean counterpart of the integral motivic action conjecture and is presented as an analogue of the Prasanna--Venkatesh conjecture.