Exoo's conjecture on the chromatic number of interval distance graphs

Let $G_D$ be the distance graph on the plane whose vertices are the points of $\mathbb{R}^2$, with two points adjacent when their distance lies in $D$. For $b>1$, write $G_{[1,b]}$ for the graph with distance set $[1,b]$, and let $\chi$ denote chromatic number.

Exoo's conjecture. For $b>1$ sufficiently close to $1$, it holds

$$\chi(G_{[1,b]})=7.$$

This conjecture strengthens the known result that $\chi(G_{[1,b]})=7$ for $b\in(\sqrt{43}/5,\sqrt{7}/2]$ and proposes that the same value persists throughout some right neighborhood of $1$. The source does not provide evidence of a resolution.