Uniqueness conjecture for metric spaces with discrete distance sets

Let $X$ be a metric space, and let $\operatorname{dist} X$ denote the set of distances between pairs of points of $X$. The set $\operatorname{dist} X$ is closed and discrete if it is a closed discrete subset of $\mathbb{R}$. Let $\operatorname{\mathcal{G\!H}}$ be the class of metric spaces under consideration for the Gromov--Hausdorff distance.

Uniqueness conjecture. Every metric space $X$ with a closed and discrete set $\operatorname{dist} X$ has the property of uniqueness in the class $\operatorname{\mathcal{G\!H}}$.

This asserts that vanishing Gromov--Hausdorff distance determines $X$ uniquely up to isometry within the stated class. The supplied text does not indicate whether this claim is proved or remains open.