The cuspidal cohomology decomposition conjecture for $GL_r$

Fix a cocompact lattice $J\subset Z(\mathbbA)/Z(F)$ and an admissible $n$-tuple $\overline{\omega}$. Let

$$H^0_{\operatorname{cusp},J}(\overline{\omega})$$

be the cuspidal intersection cohomology representation, and let $\rho_{\pi,\overline{\omega}}$ denote the representation associated with a cuspidal representation $\pi$ as in the introduction. Cuspidal cohomology decomposition conjecture. If $G=GL_r$, then, as a representation of $GL_r(\mathbbA)\times\Gamma_{F^{(n)}}$,

$$H^0_{\operatorname{cusp},J}(\overline{\omega})\simeq\bigoplus_{\substack{\pi\ \text{cuspidal representation of }GL_r(\mathbbA)\\ \pi(J)=\operatorname{Id}}}\left(\pi\boxtimes\rho_{\pi,\overline{\omega}}\right).$$

Here $\overline{\omega}$ is identified with the corresponding representation of $(\widehat{G})^n$. This is an expected automorphic–Galois decomposition of cuspidal cohomology; the supplied source does not state whether it has been proved or remains open.