Let $(X_i)$ be a countable collection of bounded metric spaces, pre-compact in the Gromov-Hausdorff metric. It is well-known that for any choice of non-principal ultrafilter $U$ on $\mathbb N$, the metric ultra-limit of $(X_i)$-s with respect to $U$ is a Gromov-Hausdorff limit of some subsequence $(X_{i_k})$.

Let $Y$ be a Gromov-Hausdorff limit of a subsequnce $(X_{i_j})$. Is it isometric to the metric ultralimit of $(X_i)$ for some choice of ultrafilter on $\mathbb N$?