On the stability of hyperbolicity under quantitative measure equivalence

A well-known result of Shalom says that lattices in SO$(n,1)$ are ${\mathrm{L}}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of fundamental domains from a uniform lattice to a non-uniform one is $({\mathrm{L}}^p,{\mathrm{L}}^{\mathrm{\infty }})$. Moreover, it is easy to see that the coupling is cobounded: the fundamental domain of the uniform lattice is contained in a union of finitely many translates of the fundamental domain of the non-uniform one. The purpose of this note is to prove that this statement is sharp in the following sense: if a ME-coupling from a hyperbolic group to a non-hyperbolic group is cobounded and $({\mathrm{L}}^p,{\mathrm{L}}^{\mathrm{\infty }})$, then $p$ must be less than some $p_0$ only depending on the hyperbolic group.