Perelman's bi-Lipschitz stability conjecture for Alexandrov spaces

Let

$$X_i^n\stackrel{\mathrm{GH}}{\to}X^n$$

be a non-collapsed Gromov–Hausdorff-convergent sequence of compact Alexandrov spaces of dimension $n$ with curvature bounded below by $-1$. Perelman's bi-Lipschitz stability conjecture. There exists $C>1$ such that, for every sufficiently large $i$, $X_i^n$ is $C$-bi-Lipschitz equivalent to $X^n$. The compactness of the limit implies that the diameters of the $X_i^n$ are uniformly bounded. This is a stability question for Alexandrov spaces under non-collapsed Gromov–Hausdorff convergence; the supplied text gives no resolution.