Euclidean peel restriction conjecture

Let $d$ be the metric induced on a finite subset of $\mathbb{R}^m$, let $\mathcal{I}$ be a subset of $\operatorname{peel}(d)$, and let $d|_{\mathcal{I}}$ denote the restriction of $d$ to $\mathcal{I}$. Euclidean peel restriction conjecture. If $d$ is the metric for a finite subset of $\mathbb{R}^m$ and $\mathcal{I} \subseteq \operatorname{peel}(d)$, then

$$\operatorname{peel}(d|_{\mathcal{I}})=\mathcal{I}.$$

This is proposed as a weaker version of the disproved claim and reflects the authors' observation that the equality holds for finite subsets of Euclidean space; the source provides no proof or resolution.