Finite-index finite-ends conjecture for Allen–Cahn solutions

Let $u:\mathbb{R}^n\to[-1,1]$ be a solution of the Allen–Cahn equation. A solution has finite Morse index if its Morse index, defined through the associated quadratic form, is finite. It has finitely many ends if there is some $R_0>0$ such that ${u=0}$ has finitely many connected components in $\mathbb{R}^n\setminus\overline{B_{R_0}}$.

Finite index implies finite ends. For $n\geq 3$, every finite Morse index solution has finitely many ends.

The conjecture is motivated by analogous results for minimal hypersurfaces and is stated as a long-standing problem. Its resolution status is not specified here.