Quantitative rigidity conjecture for global minimal-surface-type equations

Let $u$ be a global solution of an equation of minimal-surface type on $\mathbb{R}^n$, corresponding to a functional $A_{\Psi}$. Here $B_r$ denotes the radius-$r$ ball, $\nabla u$ the gradient, and $O(r^{\epsilon})$ the usual asymptotic bound. Quantitative rigidity conjecture. For some $\epsilon(n,\Psi)>0$,

$$\sup_{B_r}|\nabla u|=O(r^{\epsilon})\Rightarrow u\text{ is linear.}$$

This is a quantitative version of the known rigidity statement that global solutions with bounded gradient are linear, based on the De Giorgi–Nash–Moser theorem. The conjectured positive threshold remains open.