Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).

Let $\phi:M \to N$ be distance-preserving and surjective. Is it a Riemannian isometry? (In particular, is it smooth?)

If it helps, we can assume $\phi$ maps boundary onto boundary and interior onto interior.

When looking at the proof of Myers-steenrod (for example here), part of the problem seems to be that initial conditions do not determine a unique geodesic, if the starting point is on the boundary.

The basic idea of the proof is to express $\phi$ in exponential coordinates, and then show that the exponential representation is linear, hence smooth. However, constructing this representation relies on the uniqueness of geodesics.