Rational ellipticity of $G$-manifolds from their quotients

We prove that if a compact, simply connected Riemannian $G$-manifold $M$ has orbit space $M\mathrm{/}G$ isometric to some other quotient $N\mathrm{/}H$ with $N$ having zero topological entropy, then $M$ is rationally elliptic. This result, which generalizes most conditions on rational ellipticity, is a particular case of a more general result involving manifold submetries.