Rigidity at infinity without energy minimality in all dimensions

Establish that every energy-minimizing or non-energy-minimizing harmonic map from Euclidean space \(\mathbb{R}^n\) to the round sphere \(S^{n-1}\), whose rescalings admit the radial projection \(x\mapsto x/|x|\) as a blow-down limit, is a translate of the radial projection for every \(n\geqslant3\).

Background

The paper proves a rigidity-at-infinity theorem for dimensions $3\leqslant n\leqslant7$: an energy-minimizing map from $\mathbb{R}^n$ to $S^{n-1}$ that converges along a sequence of rescalings to the radial projection must itself be a translate of that projection. The stated conjecture asks for two extensions of this result: validity in every dimension $n\geqslant3$, and removal of the energy-minimizing assumption.