Radiality conjecture for solutions of the coupled generalized Jang equation and zero divergence system

Let $(M,g,k)$ be a spherically symmetric initial data set, and consider solutions $(f,\phi)$ of the coupled system

$$\left(g^{ij}-\frac{\phi^2f^if^j}{1+\phi^2|\nabla f|^2}\right)\left(\frac{\phi\nabla_{ij}f+\phi_if_j+\phi_jf_i}{\sqrt{1+\phi^2|\nabla f|^2}}-k_{ij}\right)=0,$$ $$\overline{\operatorname{div}}(\phi q)=0,$$

with the appropriate asymptotics for application to the Penrose inequality. Radiality conjecture. If such a spherically symmetric initial data set possesses solutions, at least some of them must be radial. Consequently, the spherically symmetric initial data set supplied by the main theorem possesses no such solutions, and this coupled-system approach cannot prove the Penrose inequality. The conjecture is presented as an expected future result; its non-radial case remains unresolved, while the cited theorem rules out smooth radial solutions for the specified data.