Entropy rigidity for foliations by strictly convex projective manifolds

Let N be a compact manifold with a foliation FN whose leaves are compact strictly convex projective manifolds. Let M be a compact manifold with a foliation FM whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to 3. Suppose we have a foliation-preserving homeomorphism f: (N, FN) → (M, FM) which is C1-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies h(N, FN) and h(M, FM) and it holds h(M, FM) ≤ h(N, FN). Additionally, if equality holds, then the leaves must be homothetic.