Nonlocal De Giorgi conjecture for the fractional Allen–Cahn equation

Let $\alpha\in(0,2)$ and let $u\in C^2(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$ be a bounded global solution of the fractional Allen–Cahn equation

$$-(-\Delta)^{\frac{\alpha}{2}}u=u-u^3.$$

Assume that

$$\partial_n u(x)>0\qquad\text{for every }x\in\mathbb{R}^n.$$

Nonlocal De Giorgi conjecture. Is $u$ one-dimensional, at least under suitable restrictions on $\alpha$ and $n$? This is the fractional analogue of De Giorgi's one-dimensional symmetry conjecture and asks whether monotone bounded entire solutions have planar level sets. The source presents it as an open question and does not specify the suitable restrictions on $\alpha$ and $n$.