If $(\mathcal{X}_{t_{i}},g_{i})$ converges to $(X_{\infty},g_{\infty})$ and $(B_{i},d_{i})$ is non-empty for large $i$ and converges to $(B_{\infty},d_{\infty})$, then $B_{\infty}$ and $X_{\infty}$ are isometric up to scaling. Furthermore, there is a subspace $B_{\infty,0}\subseteq B_{\infty}$ with $\Gamma:=B_{\infty}\setminus B_{\infty,0}$ of Hausdorff codimension 2 in $B_{\infty}$ such that $B_{\infty,0}$ is a Monge-Ampère manifold, with the Monge-Ampère metric inducing $d_{\infty}$ on $B_{\infty,0}$.