Compatible-systems conjecture for l-adic representations

A weakly compatible system $\mathcal{R}=\{R_{l,\iota}\}$ consists of semisimple l-adic representations with common Hodge–Tate data and compatible unramified Weil–Deligne realizations. It is strongly compatible when the Weil–Deligne realization exists at every prime, irreducible when every $R_{l,\iota}$ is irreducible, and geometric when it is realized in the cohomology of a smooth projective variety.

Compatible-systems conjecture. The following should hold: (1) every continuous semisimple de Rham representation $R:G_{\mathbb{Q}}\to GL_n(\overline{\mathbb{Q}}_l)$ unramified at all but finitely many primes is part of a weakly compatible system; (2) every weakly compatible system is strongly compatible; and (3) every irreducible weakly compatible system $\mathcal{R}$ is geometric and pure of weight

$$\frac{2}{\dim \mathcal{R}}\sum_{h\in\operatorname{HT}(\mathcal{R})}h.$$

This packages expected variation in $l$, local compatibility, and geometricity. The source presents it as a consequence of Fontaine–Mazur and the preceding motivic conjecture; no general resolution is stated.