Archimedean motivic action conjecture for Shimura varieties

Let $\Pi=\Pi_f\otimes\Pi_\infty$ satisfy the paper's Assumption, with $\Pi_\infty$ a non-discrete-series representation that is not a discrete series. Set

$$\mathcal{M}=H_M^1((\operatorname{Ad}\Pi)_{\mathcal{O}_E},\overline{\mathbb{Q}}(1)),\qquad \mathcal{H}^i=H^i(X)[\Pi_f].$$

Regard both as $\overline{\mathbb{Q}}$-vector spaces with the Hermitian bilinear forms induced by a fixed admissible Hermitian bilinear form on $\mathfrak{g}_{\mathbb{C}}$. Archimedean motivic action conjecture. There is an isomorphism of graded Hermitian $\overline{\mathbb{Q}}$-vector spaces

$$\bigwedge^{*}\mathcal{M}^{*}\otimes\mathcal{H}^{i_{\min}}\cong \bigoplus_{i=i_{\min}}^{i_{\max}}\mathcal{H}^{i},$$

where $i_{\min}$ and $i_{\max}$ are respectively the bottom and top degrees in which $\Pi_f$ appears in $H^*(X)$. The conjecture predicts that motivic cohomology accounts for the multiple coherent-cohomology appearances of one finite automorphic representation.