Density of mixing identity-plus-bilateral-shift perturbations

Determine whether the set of mixing operators \(\{I+B_w\in\mathcal{M}: w\in c_0(\mathbb{Z})\}\) is dense in \(\mathcal{M}\) endowed with the norm topology, where \(\mathcal{M}=\{I+B_w\in\mathcal{L}(\ell_p(\mathbb{Z})):w\in c_0(\mathbb{Z})\}\) and \(1<p<\infty\).

Background

For the norm-topologized family of identity-plus-bilateral-weighted-backward-shift operators, the paper establishes that weak mixing is typical, topological ergodicity is meager, and weak disjointness from a fixed hypercyclic operator is residual.

The paper does not determine whether the stronger property of mixing is dense in this concrete family. The unresolved question asks specifically about norm-density of mixing operators among all perturbations $I+B_w$ with weights in $c_0(\mathbb{Z})$.