Amenability characterized by non-Hausdorff first cohomology

Let $G$ be an infinite, finitely generated group. Let $M$ be a Banach space with norm $||\ ||_M$ that is also a left $\C G$-module, and suppose that $G$ acts on $M$ by continuous $\C$-linear transformations. Amenability conjecture. $G$ is amenable if and only if $H^1(G,M)$ is not Hausdorff. This proposes a characterization of amenability through the topological behavior of first cohomology for such Banach $G$-modules; the supplied text gives no resolution status or further evidence.