Alexandrov-space bi-Lipschitz embedding conjecture

Let $D>0$. Let $A$ be a bounded subset with $\operatorname{diam}(A)\leq D$ of an $n$-dimensional complete Alexandrov space $X^n$ with curvature $K\geq -1$. Alexandrov-space embedding conjecture. Every such $A$ admits a bi-Lipschitz embedding

$$f\colon A\to\mathbb{R}^N$$

with distortion less than $C(D,n)$ and image dimension $N\leq N(D,n)$. The conjecture would remove the upper curvature-bound restriction from the paper's embedding results; it is known in the volume-non-collapsed case, while the general case remains open.