Brown's uniqueness conjecture for the Calderón problem with $W^{1,n}$ conductivities

Let $n \geq 3$ and let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain. Suppose that $\gamma_1,\gamma_2 \in W^{1,n}(\Omega)$ are uniformly elliptic, and let $\Lambda_{\gamma_i}$ denote the corresponding Dirichlet-to-Neumann maps. Brown's conjecture.

$$\Lambda_{\gamma_1}=\Lambda_{\gamma_2}\quad\Longleftrightarrow\quad\gamma_1=\gamma_2.$$

This asserts uniqueness for the classical Calderón problem at the borderline $W^{1,n}$ regularity. The conjecture is still open when $n \geq 5$, to the best of the authors' knowledge.