Intrinsic norm conjecture for the maximal Dirac trace space

Let $M$ be a manifold with boundary $\Sigma$, let $D$ be the Dirac operator, and let $D_{\rm max}$ denote its maximal extension. Write $R$ for the trace map, and let $\check{R}$ be the trace space $R(\operatorname{dom} D_{\rm max})$ equipped with its $\check{R}$-norm. Let $\check{H}$ be the space defined in with its $\check{H}$-norm.

Intrinsic norm conjecture. The $\check{R}$-norm on $R(\operatorname{dom} D_{\rm \max})$ is equivalent to the $\check{H}$-norm. Moreover,

$$\check{H}=R(\operatorname{dom} D_{\rm \max})$$

as vector spaces.

This conjecture proposes an intrinsic description of the maximal trace space, beyond the equivalent norm obtained using a boundary-neighbourhood trivialization. The source provides no resolution status.