Compact immersed submanifolds of the sphere with diameter less than pi

Question

Let $M^n$ be a compact immersed submanifold of the sphere $S^{n+p}$. If $K_M\ge 1$, Toponogov's maximal diameter theorem assures that the diameter of $M$ is $\le\pi$, and that the equality holds if and only if $M$ is an $n$-dimensional sphere $S^n\subset S^{n+p}$. So, my question is: If the diameter of $M$ is $<\pi$, is there an open hemisphere of $S^{n+p}$ which contains $M$?