Exoo's conjecture on the chromatic number of epsilon-unit distance graphs

Let $G_\varepsilon$ be the graph with vertex set $\mathbb{R}^2$ in which two points are adjacent when their distance $d$ satisfies $1-\varepsilon\leq d\leq 1+\varepsilon$, and let $\chi(G_\varepsilon)$ denote its chromatic number. Exoo's conjecture. For any $\varepsilon>0$,

$$\chi(G_\varepsilon)=7.$$

Exoo's computational work suggested that the exact chromatic number is $7$ for sufficiently small positive $\varepsilon$; the stated universal claim is presented as a conjectural extension of the known results, which establish exact values for some ranges and lower bounds for others.