I have a Riemannian manifold $(X,g)$ which is compact, simply connected and with sectional curvature upper bounded by $k>0$ everywhere. Let $p\in X$ be any point and $q\in Cut(p)$ the nearest cut point (i.e. $d_g(p,q)=d_g(p,Cut(p))$), so I know that $q$ is a conjugate point or I have a closed geodesic $\gamma$ respect to which $p$ and $q$ are 'opposite' (i.e. $d_g(p,q)=\frac{1}{2}length(\gamma)$).
Now, I want to show that also in the second case $q$ is a conjugate point of $p$, or in alternative that $length(\gamma)\ge \frac{2\pi}{\sqrt{k}}$. In the second case clearly it's enough to prove that if $\gamma$ is the shortest closed geodesic of $X$ then $length(\gamma)\ge \frac{2\pi}{\sqrt{k}}$ (in this case one maybe can use Klingenberg's Lemma - see the version in 'Riemannian Geometry' of Gallot-Hulin-Lafontaine, pag.158).
Is it true what I want to show? How could I do?