Infinitesimal characters for the completed cohomology of ${\mathrm{G}\mathrm{L}}_n$ over CM fields

Let $p$ be a prime, and let $F$ be a CM field containing an imaginary quadratic field in which $p$ splits. We show that the locally analytic vectors of Hecke eigenspaces in the ($p$-adic) completed cohomology of ${\mathrm{G}\mathrm{L}}_n\mathrm{/}F$, localized at a non-Eisenstein decomposed generic maximal ideal, admit infinitesimal characters determined by the Sen operators of the corresponding Galois representations, thus confirming a conjecture of Dospinescu-Paškūnas-Schraen in this case.