The PSL(4,R) Hitchin component of a closed surface group π1(S) consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of S. We prove that the leaves of the codimension-1 foliation of any such projective structure are all projectively equivalent if and only if its holonomy is Fuchsian. This implies constraints on the symmetries and shapes of these leaves. We also give an application to the topology of the non-T0 space C(RPn) of projective classes of properly convex domains in RPn. Namely, Benzécri asked in 1960 if every closed subset of C(RPn) that contains no proper nonempty closed subset is a point. Our results imply a negative resolution for n≥2.