Existence conjecture for the Jang–zero-divergence system

Let $(M,g,k)$ be asymptotically flat Cauchy data with an outermost generalized apparent horizon boundary $\Sigma$. Define

$$\bar g=g+\phi^2df^2,$$ $$h=\frac{\phi\operatorname{Hess}f+(df\otimes d\phi+d\phi\otimes df)}{(1+\phi^2|df|_g^2)^{1/2}},\qquad q=h(v,\cdot)-k(v,\cdot),$$

where $v=\phi\nabla f/(1+\phi^2|df|_g^2)^{1/2}$. Jang–zero-divergence existence conjecture. There exists a solution $(f,\phi)$ satisfying

$$\operatorname{tr}_{\bar g}(h-k)=0,\qquad \overline{\operatorname{div}}(\phi q)=0,$$

with $\lim_{x\to\infty}f(x)=0$, $\phi^2|\nabla f|^2=o(r^{-1})$, $\nabla(\phi^2|\nabla f|^2)=o(r^{-2})$, and $\lim_{x\to\infty}\phi(x)=1$, such that $\Sigma$ has zero mean curvature in the $\bar g$ metric. The source explicitly calls this an important open problem because it implies the outermost generalized Penrose conjecture via the Riemannian Penrose inequality.