One-dimensional symmetry conjecture for the generalized fractional Allen–Cahn equation

Let $a\in(-1,1)$ and let $u$ be a bounded, smooth solution of the generalized fractional Allen–Cahn equation

$$\mathcal{L}_a u=u-u^3 \qquad \text{in }\mathbb{R}^n.$$

Assume that $\partial_{x_n}u>0$. Generalized fractional De Giorgi conjecture. If $n$ is sufficiently small, then $u$ is one-dimensional. This is the natural analogue of the fractional De Giorgi conjecture for the operator $\mathcal{L}_a$ arising in the water-wave framework. The surrounding discussion does not provide a resolution or a dimension threshold, so the conjecture remains open.