Low-h nonattainment conjecture for Néel-wall energies at winding number d minus alpha over pi

Let $d\in\mathbb{N}\setminus\{1\}$, and let $\mathcal{E}_h(w)$ denote the infimum of the energy among profiles with winding number $w$. Low-$h$ nonattainment conjecture. There exists $K\in(0,1)$ such that $\mathcal{E}_h(d-\alpha/\pi)$ is not attained for every $h\in[0,K]$. This complements the conjectured attainment near $h=1$ and excludes $d=1$, for which the claim is not asserted.