Finite $\epsilon$-unit distance graphs

In 2005, Exoo posed the following question. Fix $\epsilon$ with $0\leq\epsilon<1$. Let $G_\epsilon$ be graph whose vertex set is Euclidean plane, where two vertices are adjacent iff distance between them lies in closed interval $[1-\epsilon,1+\epsilon]$. What chromatic number $\chi(G_\epsilon)$ of this graph? The case $\epsilon=0$ precisely classical ``chromatic plane'' problem. a 2018 preprint, de Grey shows that $\chi(G_0)\geq 5$; proof relies heavily on machine computation. 2016, Grytczuk et al. proved weaker result human-comprehensible but nonconstructive proof: whenever $0<\epsilon<1$, we have $\chi(G_\epsilon)\geq 5$. (This lower bound $5$ was later improved by Currie and Eggleton to $6$.) De Bruijn - Erd\H{o}s theorem (which axiom choice) then guarantees existence, for each $\epsilon$, finite subgraph $H_\epsilon$ such $\chi(H_\epsilon)\geq paper, explicitly construct graphs $H_\epsilon$. We find needed create no more than $2\pi(15+14\epsilon^{-1})^2$. Our can done hand without aid computer.