Minimal-interface characterization conjecture for the relaxed functional

Let $\rho(z,\phi)$ be the relaxed functional for an admissible pair $(z,\phi)$, and define

$$oindent\Phi_+ = \{x\in\Omega:\phi(x)>0\},\qquad \Phi_- = \{x\in\Omega:\phi(x)<0\},\qquad C\Phi=\Omega\backslash(\Phi_+\cup\Phi_-).$$

If $\lambda^n(C\Phi)>0$, then $z$ is not uniquely identified and may attain every value in $[0,1]$ on $C\Phi$. Minimal-interface characterization conjecture. The infimum over admissible $z$ satisfies

$$\inf_{z\text{ admissible}}\rho(z,\phi)=2\beta\mathcal{H}^{n-1}(S)+\|\phi-\phi_0\|_{H^1(\Omega)}^2,$$

where $S\subset C\Phi$ is a surface of minimal $(n-1)$-dimensional Hausdorff measure that divides $\Omega$ into two sets, one completely containing $\Phi_+$ and the other containing $\Phi_-$. The characterization describes the relaxed functional through a minimal separating interface in the zero level-set region; the source gives no resolution of this conjecture.