*No*.

It is clear that this is true for a sphere, but cutting of a spherical cap from the sphere leaves the intersection points with the planes the same, but now there is a unique shortest path through this new flat area.

$\qquad\qquad\qquad\qquad\quad$ <img src="https://i.sstatic.net/CCVbS.png" width="280" />

Of course you can make this object strictly convex, differentiable etc. by smoothing out the cut.