Entropy rigidity for foliations by strictly convex projective manifolds

Let N be a compact manifold with a foliation F-N whose leaves are compact strictly convex projective manifolds. Let M be a compact manifold with a foliation F-M whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to 3. Suppose we have a foliation-preserving homeomorphism f : (N, F-N) -> ( M, F-M) which is C-1-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies h( N, F-N) and h( M, F-M) and it holds h(M, F-M) <= h( N, F-N). Additionally, if equality holds, then the leaves must be homothetic.