Let $M$ be a complete connected Riemannian manifold. Fix $p \in M$.
Assume every point in $M$ has a unique minimizing geodesic connecting it to $p$. Is it true that for every point, the only geodesic connecting it to $p$ is the minimizing one?
(Does the answer change if we count periodic geodesic as one?)
Remarks:
(1) In all the examples I have checked so far this holds. Of course, if for some $q \in M$ there is a unique minimizing geodesic, it does not imply there is a unique geodesic from $p$ to $q$ (think of a circle).
For spheres,tori,cylinders - the claim vacuously holds, since there is no point satisfying the hypothesis (every point has an "antipodal" point where there is more than one minimizing geodesic).
(2) If the assertion is true, it implies something quite strong on manifolds where minimizing geodesics are always unique; By this answer such a manifold must be diffeomorphic to $\mathbb{R}^n$ (via at the exponential map).
Actually, one can adapt the argument of that answer to see that if there is one such point $p$ (that minimizing geodesics from it to all other points are unique), then the manifold will be diffeomorphic to $\mathbb{R}^n$.
Thus, to refute the conjecture, it is enough to find a manifold with one such point which is not diffeomorphic to $\mathbb{R}^n$.
According to section 4 of this paper:
Stephanie B. Alexander, I. David Berg, and Richard L. Bishop, Cut loci, minimizers, and wavefronts in Riemannian manifolds with boundary, Michigan Math Journal, Volume 40, Issue 2 (1993), 229-237
for a connected complete Riemannian manifold $(M,g)$ without boundary the cut-locus $C(p)$ of a point $p\in M$ equals the closure of the set $N(p)$ consisting of points $q\in M$ such that there is more than one minimizing geodesic between $p$ and $q$.
Now, your assumption is that $N(p)=\emptyset$ for some $p\in M$. Hence, $C(p)=\emptyset$ for this $p\in M$. Thus, according to the answer to your earlier question (I think you asked this question several times, I remember writing an answer), it follows that any point in $M$ is connected to $p$ by a unique geodesic as $\exp_p$ is a diffeomorphism.