Shalom's conjecture on uniformly Lipschitz affine actions of hyperbolic groups

Let $G$ be a Gromov hyperbolic group, and consider affine actions of $G$ on a Hilbert space whose linear parts are uniformly bounded representations. An affine action is proper when its orbit maps are proper, and uniformly Lipschitz when the action maps have uniformly bounded Lipschitz constants. Shalom's conjecture. Once the requirement that the linear part be unitary is dropped, every Gromov hyperbolic group, including those having Property (T), admits a proper uniformly Lipschitz affine action on a Hilbert space. The conjecture predicts that allowing uniformly bounded, rather than unitary, linear parts removes the obstruction to proper affine actions even for hyperbolic groups with Property (T); its resolution is not indicated here.