Motivic factorisation of the derived Hecke action

Let $Y(K)$ be the arithmetic manifold, let $E$ be the coefficient field, and let $M_{\mathrm{coad}}$ be the coadjoint motive with its integral model $(M_{\mathrm{coad}})_\mathbb Z$. The comparison map from the torus-derived Hecke operators to motivic cohomology is the map described in the source. Motivic factorisation conjecture. The derived Hecke algebra action on $H^*(Y(K),E)$ factors through motivic cohomology. Moreover, the resulting action of

$$\operatorname{Hom}_{\mathbb Q}\left(H^1_{\mathcal M}\left((M_{\mathrm{coad}})_\mathbb Z,\mathbb Q(1)\right),\mathbb Q\right)\otimes_{\mathbb Q}E$$

on $H^*(Y(K),E)$ comes from a rational action of

$$\operatorname{Hom}_{\mathbb Q}\left(H^1_{\mathcal M}\left((M_{\mathrm{coad}})_\mathbb Z,\mathbb Q(1)\right),\mathbb Q\right)$$

on $H^*(Y(K),\mathbb Q)$. This predicts that the derived Hecke action has a motivic and rational origin.