Half-space rigidity conjecture for semilinear solutions in dimension three

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$, and let $u$ solve

$$\Delta u=f(u)\quad\text{in }\mathbb{R}^3.$$

Assume that $u$ has finitely many critical points and that $u(x)>0$ implies $x_1>0$. Half-space rigidity conjecture. Then $u$ is one-dimensional in the sense of the one-dimensional rigidity conjecture; equivalently, after a rigid motion it has the corresponding one-dimensional profile.

This is a semilinear analogue of half-space rigidity results for free-boundary problems and minimal surfaces. The source cites related results but does not state that this conjecture has been resolved.