Let $\{X_i^n\}$ be a sequence of smooth compact Riemannian $n$-dimensional manifolds with boundary. Assume that this sequence has uniformy bounded below sectional curvature, and each $X_i$ is geodesically locally convex near its boundary (the latter assumption is equivalent that $X_i$ is an Alexandrov space). Assume $\{X_i\}$ converges in the Gromov-Hausdorff sense to a compact smooth Riemannian manifold $X$ with boundary and $\dim X=n$, i.e. there is no collapse.

Is it true that volume of the bondary $vol_{n-1}(\partial X_i)$ converges to $vol_{n-1}(\partial X)$? The case $n=2$ is already interesting to me.

**Remarks.** (1) By the Perelman stability theorem $X_i$ is homeomorphic to $X$ for $i\gg 1$.

(2) By Burago-Gromov-Perelman one has convergence of volumes of manifolds $vol_n(X_i)\to vol_n(X)$.