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.