Let $(M, g)$ be a complete Riemannin manifold and suppose $p, q \in M$ re points s.t. $d_g(p, q)$ is equal to the distance from $p$ to its cut locus, with $q \in \text{Cut}(p)$. If $q$ is not conjugate to $p$ along some minimizing geodesic segment, does it follow that there exist exactly two minimizing geodesic segments $\gamma_1, \gamma_2 : [0, b] \to M$ from $p$ to $q$ s.t. $\gamma_1'(b) = - \gamma_2'(b)$?
My first thought was to make a closed loop going from $p$ to $q$ along $\gamma_1$ and then from $q$ to $p$ along $-\gamma_2$. Then I wanted to deform this loop into one of the geodesics to obtain a Jacobi field vanishing at $q$, but I fail to see how each curve would be a geodesic, or how this depends on $\gamma_1'(b) \neq -\gamma_2'(b)$. Then I started wondering if there was something missing from the problem. If I reverse the direction of $\gamma_2$ and concatenate both geodesics, I would obtain a closed geodesic. This seems to imply that on any complete manifold where the closest point on the cut locus of $p$ to $p$ is not conjugate to $p$, we would have closed geodesics. Is this even true?