Fibred cofinitely-coarse embeddability of box families and proper isometric affine actions on uniformly convex Banach spaces

In this paper we show that a countable, residually amenable group admits a proper isometric affine action on some uniformly convex Banach space if and only if one (or equivalently, all) of its box families admits a fibred cofinitely-coarse embedding into some uniformly convex Banach space.