Conformal-factorization conjecture for normalized symmetric gradients

Let $\Omega\subset \mathbb R^n$ be a bounded open connected domain with $n\geq 3$. Let $u\in W^{1,n}(\Omega)$ and $v\in W^{1,1}(\Omega)$ satisfy

$$\det(\nabla u)>0,\qquad \det(\nabla v)>0$$

almost everywhere, and suppose that

$$\int_{\Omega}\left(\frac{\|\nabla u\|^n}{\det(\nabla u)}\right)^p dz<\infty$$

for some $p>n-1$. If their normalized symmetric gradients agree almost everywhere,

$$\frac{S(\nabla u(z))}{|S(\nabla u(z))|}=\frac{S(\nabla v(z))}{|S(\nabla v(z))|}\quad\text{for almost every }z\in\Omega,$$

then there exists a Möbius transformation $\Phi$ such that $v=\Phi\circ u$. Normalized symmetric-gradient conjecture. Under these hypotheses, there exists a Möbius transformation $\Phi$ satisfying $v=\Phi\circ u$. The conjecture seeks a higher-dimensional analogue of conformal factorization, extending the planar relationship between mappings with matching geometric distortion.