Suppose $(M,g)$ is a complete, connected Riemannian manifold, $p \in M$, and $\mathrm{Cut}(p)$ denotes the cut locus of $p$, i.e. the set of all points $q \in M$ so that there is a minimizing geodesic from $p$ to $q$ that ceases to be minimizing past $q$. I'm trying to prove the following (Problem 10-23 in John Lee's "Introduction to Riemannian Manifolds"):
Suppose $q \in \mathrm{Cut}(p)$ satisfies $d_g(p,q) = d_g(p,\mathrm{Cut}(p)) = \mathrm{inj}(p) =: b$. Then either $q$ is conjugate to $p$ along some minimizing geodesic segment, or there are exactly two unit-speed geodesic segments $\gamma_0, \gamma_1 : [0,b] \to M$ such that $\dot\gamma_0(b) = -\dot\gamma_1(b)$.
By "$p$ is conjugate to $q$", we mean there is a Jacobi field $J$ along a geodesic from $p$ to $q$ with $J(p) = 0$ and $J(q) = 0$; or equivalently, there is a proper smooth variation through geodesics from $p$ to $q$. Furthermore, $\mathrm{inj}(p)$ denotes the injectivity radius at $p$.
What I've tried: By a previous result, assuming $q$ is not conjugate to $p$ along any minimizing geodesic, there are at least two minimizing unit-speed geodesics $\gamma_0$ and $\gamma_1$ from $p$ to $q$. Suppose $\dot\gamma_0(b) \neq -\dot\gamma_1(b)$. Let $\gamma : [0,2b] \to M$ be the closed loop based at $p$ given by $\gamma_0$ and (the reversed parametrization of) $\gamma_1$, and let $\Gamma : [0,\delta] \times[0,2b] \to M$ be a smooth variation along $\gamma$ through unit-speed curves that coincide with $\gamma_0$ and $\gamma_1$ outside of an $\epsilon$-neighborhood of $q$, and inside this $\epsilon$-neighborhood of $q$, the variation field $V$ satisfies $V(b) = -\left(\dot\gamma_0(b) + \dot\gamma_1(b)\right)$. Assume $\Gamma_0 = \gamma$. (Essentially, we are slightly shortening and contracting the loop $\gamma$ by "rounding the corner" at $q$.)
One can show there is a small $s \in (0,\delta]$ such that $L_g(\Gamma_s) < L_g(\gamma) = 2b$. Suppose the image of $\Gamma_s$ differs from the image of $\gamma$ on the interval $[b-\epsilon, a]$ for some $a \in (b-\epsilon, b+\epsilon]$. Let $\sigma = \Gamma_s|_{[b-\epsilon,a]}$. I've been able to show that $\sigma$ does not intersect $\mathrm{Cut}(p)$. So $\sigma$ is a smooth curve in $M$ within the injectivity radius of $p$. Because $\exp_p$ is a diffeomorphism near $0$ onto its image, and that image includes $\sigma$, we can project $\sigma$ to a curve $\tilde\sigma$ on the unit sphere in the tangent space $T_pM$. So let $\Sigma : [b-\delta, a] \times [0,b] \to M$ be the variation given by $$ \Sigma(s,t) = \exp_p\left(t\tilde\sigma(s)\right). $$ Then $\Sigma$ is a variation through unit-speed geodesics that start at $p$.
My question: Can we show $\Sigma$ is a proper variation? If so, this would contradict our assumption that $q$ is not conjugate to $p$ along any minimizing geodesic.
I know that by minimality, the first time each curve $\Sigma_s : [0,b] \to M$ can possibly intersect either $\gamma_0$ or $\gamma_1$ is at $t=b$. Also, the curve $\Sigma(\cdot, b) : [b-\delta, a] \to M$ is a closed curve based at $q$. So it suffices to show this curve is constant. Is there an obvious reason why this should be the case?
EDIT: See comments for a solution sketch of the original problem. Constructing a proper variation through geodesics, as I was trying to do, turns out not to be the most straightforward approach; it's better to use the fact that $q$ is a regular value of $\exp_p : T_p M \to M$ if we assume $q$ is not conjugate to $p$.