I am wondering if the following statement is true or not:

Let $ M $ be a complete Riemannian manifold with sectional curvature $ K $. Let $ \tilde{M} $ be the simply connected complete Riemannian manifold of constant curvature $ \kappa $ with $ K\geq \kappa $ and $ \dim M = \dim \tilde{M} = n $. Let $ p,q \in M $ and $ \gamma $ a shortest geodesic from $ p $ to $ q $ with $ L(\gamma) = r $ i.e. dist$ (p,q) = r $. Does it hold that for arbitrary $\tilde{p} \in \tilde{M} $, there exists $\tilde{q} \in \tilde{M} $ with dist$ ( \tilde{p} , \tilde{q} ) \geq r$? thanks a lot