Let $(M^n,g_i)$ be a sequence of smooth complete Riemannian manifold with $|sec_{g_i}| le 1$. Suppose $(M_i^n,g_i)$ converges to a limit space $(X^{n-1},d)$ in the Gromov-Hausdorff sense, where the Hausdoff dimension of $X$ is $n-1$.

Can we show that $X$ contains no boundary point? Here, a point is a boundary point of $X$ if its tangent cone is isometric to $mathbb R^{n-2} times mathbb R_+$.