This question stems from the discussion in:

how to define the injectivity radius of manifolds with boundary?

Suppose $(M,g)$ is a compact Riemannian manifold with boundary. In this context, let the injectivity radius of a point $x$ be the minimum distance from $x$ at which there is a point $y$ with more than one length-minimizing geodesic connecting $x$ to $y$.

**Is it true that the injectivity radius as defined this way is bounded below by some nonzero value? If so, is there a standard reference for this fact?**

In the discussion linked above, the following theorem is referenced:

Corollary 2. If for a complete Riemannian manifold with boundary, M, the sectional curvatures of the interior and the outward sectional curvatures of the boundary are no greater than $K$, then $N(p,\frac{\pi}{2K})$ is open in M and the distance function from p is convex on $N(p,\frac{\pi}{2K})$.

Where $N(p,\frac{\pi}{2K})$ is the set of points connected to $p$ by a unique geodesic of length $\frac{\pi}{2K}$ or less.

(Reference Paper) https://www.ams.org/journals/tran/1993-339-02/S0002-9947-1993-1113693-1/S0002-9947-1993-1113693-1.pdf

This seems like it is close to the result I am looking for. Earlier in the paper, it is also stated that there are no conjugate points in $N(p,\frac{\pi}{K})$.

Is there a simple step from this result that proves that the injectivity radius is nonzero?