Establish motivicity of all algebraic Hecke characters

Establish that every algebraic Hecke character is motivic, meaning that its $3$-adic realization arises from a factor of the cohomology of a motive.

Background

The paper defines a Hecke character as motivic when its $3$-adic character occurs in the realization of a motivic factor after base change. This property is needed to connect Hecke-character periods with the motivic period formalism used throughout the paper. The authors use motivicity for the specific Jacobi-sum characters arising in their examples, but state the general assertion only as a conjectural expectation.