Let $X$ be a proper geodesic space which is uniquely geodesic. Let $\phi:[0,1]\times[0,1] \to X$ be a function which satisfies the following:
The maps $\phi(0,\cdot)$, $\phi(\cdot,0)$, $\phi(1,\cdot)$, and $\phi(\cdot,1)$ are all (linearly parametrized) geodesics. Furthermore, for each fixed $s$, the map $\phi(s,\cdot)$ is a (linearly parametrized) geodesic connecting $\phi(s,0)$ to $\phi(s,1)$.
Given the above conditions, is it true that that for any fixed $t$, the map $\phi(\cdot,t)$ is a geodesic connecting $\phi(0,t)$ to $\phi(1,t)$? If not, is there a condition we can apply for which this is true (e.g. the space must be Hadamard)?