Let $\{K_i\}$ be a sequence of convex compact $n$-dimensional subsets in a Euclidean space $\mathbb{R}^n$. Assume $\{K_i\}$ converges in the Hausdorff metric to a convex compact set $K$ which is also $n$-dimensional.

Consider the sequence of boundaries $\{\partial K_i\}$ equipped with the induced intrinsic (!) metrics.

Is it true that the sequence $\{\partial K_i\}$ with these intrinsic metrics converges to $\partial K$ (again, equipped with the intrinsic metric) in the Gromov-Hausdorff sense?

A proof or a reference will be helpful.