Finite-index De Giorgi conjecture in dimensions four through seven

Let $u:\mathbb{R}^n\to[-1,1]$ solve the Allen–Cahn equation and have finite Morse index.

Finite index De Giorgi conjecture. If $4\leq n\leq 7$, then $u$ is one-dimensional.

The paper proves the analogous classification in $\mathbb{R}^4$ under a bounded energy-density hypothesis and obtains the result conditionally in dimensions $4\leq n\leq7$ on the classification of stable solutions. The unconditional conjecture is therefore presented as open in the supplied text.