Equivalence of the induced norm with the Sobolev norm

Let $\Omega\subset\mathbb{R}^3$ be a Lipschitz domain, let $\emptyset\neq\Gamma\subset\partial\Omega$ be relatively open, and let $P\in\mathsf{L}^{\infty}_{}(\Omega;\mathbb{R}^{3\times3})$ satisfy $\det P\geq c^+>0$. Define

$$|||u|||=\left\|\operatorname{sym}(\nabla uP^{-1})\right\|_{\mathsf{L}^{2}_{}(\Omega)}$$

and let $\mathsf{H}^{1}_{\circ}(\Omega,\Gamma)$ and the completion under $|||\cdot|||$ be the corresponding completions of $\mathsf{C}^{\infty}_{\circ}(\Omega,\Gamma)$. Norm-equivalence conjecture. The induced norm and the $\mathsf{H}^{1}_{}(\Omega)$ norm should be equivalent when $\operatorname{Curl}P\in\mathsf{L}^{p}_{}(\Omega;\mathbb{R}^{3\times3})$ and $\det P\geq c^+>0$ for some $p>1$, or even for $p\geq1$. The source notes that equivalence is known when $P\in\mathsf{C}^{0}_{}(\overline{\Omega})$ with $\det P\geq c^+>0$, while the asserted lower-regularity extension remains open.