Let $M^n$ be an $n$-dimensional smooth compact Riemannian manifold with boundary. Assume that the sectional curvature of $M$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form of the boundary is at least $\lambda$.

Question. Does there exist a lower bound on the sectional curvature of the boundary $\partial M$ in terms of $n,\kappa,\lambda$ and (possibly) $D$?