The classification of d-cube densities above 5/8

Let $H$ be a configuration in $Q_d$, and let $\lambda(H,d)$ denote its $d$-cube density. Classification conjecture above $5/8$. If

$$\frac{5}{8}<\lambda(H,d)<1,$$

then either $H$ is layered and $\lambda(H,d)=\frac{2}{3}$, or $d=3$, $H=W_2$, and $\lambda(H,d)=\frac{3}{4}$. The claim would classify all nontrivial $d$-cube densities in the interval above $5/8$; the paper notes that layered configurations with density $2/3$ exist for every $d\geq 3$, while $W_2$ provides the stated 3-dimensional example. What remains open is whether these are the only possibilities.