Existence conjecture for global minimal-surface-type solutions asymptotic to Lawson cones

Let $C_{kl}$ be the Lawson cone and let $\Phi$ be an integrand constructed as in Theorem Main. An elliptic extension of $\Phi$ is an extension to $\mathbb{R}^{k+l+3}$ that remains elliptic. Existence conjecture. For any integrand $\Phi$ as constructed in Theorem Main, there exists an elliptic extension of $\Phi$ to $\mathbb{R}^{k+l+3}$ and a nonlinear global solution of the corresponding equation of minimal-surface type in $\mathbb{R}^{k+l+2}$ whose graph is asymptotic to $C_{kl} \times \mathbb{R}$. Moreover, the gradient of this solution grows at the same rate as the leaves in the foliation associated to $\Phi$ approach $C_{kl}$. The preceding discussion relates this prediction to entire minimal graphs asymptotic to minimizing cones and to the gradient growth determined by the foliation. The conjecture remains unresolved in the source.