Metric completion conjecture for the outer metric on embeddings

Let $M$ be a smooth, compact manifold without boundary with $\dim M=m$, and let $\operatorname{Emb}(M,\mathbb R^d)$ be the space of smooth embeddings equipped with the outer metric $\operatorname{dist}^{\mathcal O}$. For $s>d/2+1$, set

$$s'=s-(d-m)/2.$$

Let $\mathcal E^{s'}(M,\mathbb R^d)$ denote the space of $H^{s'}$-embeddings. Metric completion conjecture. The metric completion of $(\operatorname{Emb}(M,\mathbb R^d),\operatorname{dist}^{\mathcal O})$ is the space $\mathcal E^{s'}(M,\mathbb R^d)$.

The conjecture would identify the completion suggested by the local equivalence between the outer geodesic distance and the $H^{s'}$ norm. The key missing ingredient is an extension operator around arbitrary $H^{s'}$-embeddings; the argument currently applies only to smooth embeddings.