Universality of G-subshifts with specification

Let G be an infinite countable amenable group and let ( X , G ) be a G -subshift with specification, containing a free element. We prove that ( X , G ) is universal, i.e., has positive topological entropy and for any free ergodic (measure-preserving) G -action on a standard probability space, ( Y , ν , G ) , with h ( ν ) < h top ( X ) , there exists a shift-invariant measure μ on X such that the systems ( Y , ν , G ) and ( X , μ , G ) are isomorphic. In particular, any K -shift (consisting of the indicator functions of all maximal K -separated sets) containing a free element is universal.