Weinstein, in the article The Cut Locus and Conjugate Locus of a Riemannian Manifold proved that for a compact differentiable manifold $M$, which is not homeomorphic to $S^2$, there always exists a Riemannian metric on $M$ and a point $p\in M$ whose conjugate locus and cut locus are disjoint.
I was thinking if the following has a positive answer:
Let $M$ be a Finsler manifold which is not Riemannian. Then is it true that for every $p\in M$, the intersection of cut and conjugate locus is non-empty?