The geometric-to-strict compatibility conjecture

Let $K$ be a number field, let $G_K$ be its absolute Galois group, and let $L$ be a number field or a finite extension of a number field over which a semisimple geometric representation is defined. A weakly compatible system of Galois representations is called strictly compatible if, for every finite place $v$ of $K$, there is a Weil–Deligne representation over an algebraic closure of its coefficient field whose specializations agree with the Frobenius-semisimplified Weil–Deligne representations at every finite place of the coefficient field.

Geometric-to-strict compatibility conjecture. Any semisimple geometric representation

$$G_K\to\operatorname{GL}_n(L)$$

is part of a strictly compatible system of Galois representations.

This is presented as a consequence of the Fontaine–Mazur conjecture together with standard conjectures on the étale cohomology of algebraic varieties over number fields. The source gives no resolution or restricted case sufficient to determine the status here.