  We give new counterexamples to a question of Karsten Grove, whether there are
only finitely many rational homotopy types among simply connected manifolds
satisfying the assumptions of Gromov's Betti number theorem. Our
counterexamples are homogeneous Riemannian manifolds, in contrast to previous
ones. They consist of two families in dimensions 13 and 22. Both families are
nonnegatively curved with an additional upper curvature bound and differ
already by the ring structure of their cohomology rings with complex
coefficients. The 22-dimensional examples also admit almost nonnegative
curvature operator with respect to homogeneous metrics.
