Taylor's strict local-global compatibility conjecture for cuspidal representations of ${\mathrm{GL}}_{2n}$

Let $\Pi$ be a cuspidal automorphic form on ${\mathrm{GL}}_{2n}({\mathbb A}_{\mathbb Q})$ with infinitesimal character $\chi_H$, where $H$ is a multiset of distinct integers. A strictly compatible system of Galois representations is a collection whose Weil–Deligne realizations agree at every prime and whose Hodge–Tate weights are $H$. Taylor's strict local-global compatibility conjecture. There is a strictly compatible system of Galois representations $(\rho_{\Pi,\ell})$ associated to $\Pi$ with Hodge–Tate weights $H$ such that local-global compatibility holds for all primes. This conjecture predicts the existence of compatible Galois representations realizing the automorphic data of $\Pi$ at every place; the supplied text gives no resolution, so its status remains open.