Rigidity conjecture for the half-space Allen–Cahn linearized equation

Let $\phi: \mathbb{R}^{n-1} \times \mathbb{R}^+ \to \mathbb{R}$. Suppose that $\phi(s,0) \equiv 0$. Let $f: \mathbb{R}^{n-1} \to \mathbb{R}$, and suppose

$$[\Delta_{\mathbb{R}^n} - W”(g)](\phi) = f(s) \dot{g}(t).$$

Here $g$ is the one-dimensional Allen–Cahn heteroclinic and $w$ is the function specified by the equation referred to as $wEquation$. Rigidity conjecture. There exists $c \in \mathbb{R}$ such that $f(s) \equiv c$ and

$$\phi(s,t)=c w(t).$$

This conjecture asserts that solutions with zero boundary data and forcing proportional to the translational mode $\dot{g}(t)$ have no dependence on the tangential variable $s$. Its relevance is to the analysis of the Dirichlet-to-Neumann map for Allen–Cahn solutions near a minimal hypersurface; the source records it as future work, and no resolution is supplied here.