One-dimensional rigidity conjecture for semilinear solutions with a topological one-dimensional model

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

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

and suppose there is a homeomorphism $\Phi:\mathbb{R}^n\to\mathbb{R}^n$ and a one-dimensional solution $U$ of the free-boundary problem such that $u(x)=U(\Phi(x))$. One-dimensional rigidity conjecture. Then $u$ is one-dimensional: after a rigid motion,

$$u(x)=V(x_1),\qquad V”(s)=f(V(s)).$$

This should hold at least for $n\le3$.

The conjecture asks whether topological equivalence to a one-dimensional free-boundary profile forces genuine one-dimensionality for the semilinear equation. The source immediately raises a local version, so the global assertion is presented as unresolved.