Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the surface is locally geodesically convex.

**Is it true that the number of connected components of the boundary is bounded above by a constant depending on $D$ only?**

Remark. I am not a specialist, but if I understand correctly the answer should be positive and follow from very general results in the Alexandrov geometry. Indeed due to local convexity, $X$ is an Alexandrov space of curvature at least $-1$. Each boundary component is an extremal subset in the sense of Perelman-Petrunin. In general the number of extremal subsets of a compact $n$-dimensional Alexandrov space of curvature at least $-1$ and diameter at most $D$ is bounded above by a constant depending on $n,D$ only. **Is this argument correct? Anyway I would prefer to have a more direct argument in my case.**