Trace and extension conjecture for Sobolev embeddings

Let $M$ be a smooth, compact manifold without boundary with $\dim M=m$, and let $\mathcal E^{s'}(M,\mathbb R^d)$ be the space of $H^{s'}$-embeddings. For $s>d/2$, set

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

The trace operator

$$\operatorname{Tr}:\mathcal E^{s'}(M,\mathbb R^d)\times H^s(\mathbb R^d)\to H^{s'}(M),\qquad (q,f)\mapsto \operatorname{Tr}_q f,$$

is required to be continuous. Trace and extension conjecture. Around each $q\in\mathcal E^{s'}(M,\mathbb R^d)$ there exists an open neighborhood $\mathcal U\subseteq\mathcal E^{s'}(M,\mathbb R^d)$ and a continuous extension map

$$\operatorname{Ex}:\mathcal U\times H^{s'}(M)\to H^s(\mathbb R^d),\qquad (q,f)\mapsto \operatorname{Ex}_q f,$$

satisfying

$$\operatorname{Tr}_q\operatorname{Ex}_q f=f.$$

Such trace and extension operators would extend the smooth-embedding construction to Sobolev embeddings and provide the missing tool for studying the metric completion conjecture. The statement is presented as an open conjecture in the source.