Suppose that $(M,g)$ is a compact connected smooth Riemannian manifold without boundary. Let $U \subset M$ be a smooth submanifold of codimension zero with smooth boundary and assume that $U$ is nontrapping, meaning that all inextendible unit speed geodesics eventually exit the set $U$. Does there exist a point $p \in M\setminus U$ such that $A(p) \subset M\setminus U$?

Here, $A(p)$ is the antipodal set of $p$ defined by $$ A(p)=\{ q\in M\,:\, \textrm{dist}_g(q,p)\geq \textrm{dist}_g(y,p) \quad \forall\, y\in M\}.$$