Isotypic modularity conjecture for CM cycles

Let $\mathcal A$ be the set of cuspidal automorphic representations defined above. For $\pi\in\mathcal A$, let

$$\overline{CM}(\pi):=\varinjlim_S L_{\widetilde\pi^S}\otimes\overline{CM}[\pi^S]\subset\overline{CM}(\Omega),$$

where $S$ ranges over the relevant finite sets and $L_{\widetilde\pi^S}$ is the Hecke character on the corresponding invariant space. Define $\overline{CM}(\widetilde\pi)\subset\overline{CM}(\Omega^{-1})$ similarly. Isotypic modularity conjecture. For every $\pi\in\mathcal A$,

$$\overline{CM}(\pi)=\widetilde\pi^\infty\otimes\overline{CM}[\pi],$$

and the corresponding equality holds for $\overline{CM}(\widetilde\pi)$. This is stated as equivalent to the preceding full modularity conjecture and refines it representation by representation; the supplied source gives no resolution status.