Recently Averkov showed that various convex cones related to nonnegative polynomials do not have K-lifts (representations as projections of linear sections of K) where K is a Cartesian product of positive semidefinite cones of "small" size. In this talk I'll discuss an extension of this result that says that convex bodies with certain neighborliness properties do not have K-lifts whenever K is a Cartesian product of cones, each of which does not have any long chains of faces (such as smooth cones, low-dimensional cones, and cones defined by hyperbolic polynomials of low degree).