Nonattainment conjecture for Néel-wall energies at winding numbers d and d plus alpha over pi

Let $h\in[0,1)$, let $d\in\mathbb{N}=\{1,2,\dots\}$, and let $\mathcal{E}_h(w)$ denote the infimum of the energy among profiles with winding number $w$. Nonattainment conjecture. Neither $\mathcal{E}_h(d)$ nor $\mathcal{E}_h(d+\alpha/\pi)$ is attained. This concerns the cases not covered by the existence results; the preceding discussion identifies escape of walls to infinity as the possible source of noncompactness, but no proof is known here.