Existence conjecture for Néel-wall energies at winding number d minus alpha over pi near h equals 1

Let $d\in\mathbb{N}$, and let $\mathcal{E}_h(w)$ denote the infimum of the energy among profiles with winding number $w$. Existence conjecture. There exists $H\in(0,1)$ such that $\mathcal{E}_h(d-\alpha/\pi)$ is attained for every $h\in[H,1)$. This predicts attainment near $h=1$ in the cases left unresolved by the proved existence and nonexistence results.