Occurrence of every irreducible tensor-induction factor

Determine whether every irreducible factor of the tensor induction \(\operatorname{Ind}^{Q}_{F}\rho_\lambda\) occurs in the \(\lambda\)-isotypic component of the middle-degree completed cohomology \(\tilde{H}^{d}(K^{p})_{\overline{\mathbb{Q}}_{p}[\lambda]}\) for a strongly irreducible Galois representation \(\rho_\lambda\).

Background

The paper proves that, under strong irreducibility and a non-scalar local-inertia hypothesis, the relevant completed-cohomology eigenspace is isomorphic to a tensor induction tensored with a representation of $\mathrm{G}(\mathbb{Q}_p)$. The authors explain that semisimplicity implies only that every irreducible constituent of the eigenspace is a simple factor of the tensor induction.

The unresolved issue is whether the converse holds: namely, whether all irreducible constituents of the tensor induction necessarily appear in the completed-cohomology eigenspace. The remark specifically questions whether Nekovář’s method establishes this stronger multiplicity-support assertion.