Erschler–Tóth–Werner localization conjecture for stuck walks

Let $\alpha>1/3$, define $\alpha_1:=+\infty$ and, for $L\ge2$,

$$\alpha_L:=\frac{1}{1+2\cos(\frac{2\pi}{L+2})}.$$

For the walk, let $R'$ be the set of sites visited infinitely often,

R':=\left\\{j\in\mathbb{Z}:Z_{\infty}(j)=\infty\right\\}.

Erschler–Tóth–Werner's conjecture. If $\alpha\in(\alpha_{L+1},\alpha_L)$, then $|R'|=L+2$ almost surely. This conjecture predicts exact localization of the walk on $L+2$ sites throughout each interval $(\alpha_{L+1},\alpha_L)$; the cited preceding result establishes only that the probability of this localization is positive.