The work is devoted to one of the variations of the Hadwiger--Nelson problem on the chromatic number of the plane. In this formulation one needs to find for arbitrarily small $\varepsilon$ the least possible number of colors needed to color a Euclidean plane in such a way that any two points, the distance between which belongs to the interval $[1-\varepsilon, 1+\varepsilon]$, are colored differently. The conjecture proposed by G. Exoo in 2004, states that for arbitrary positive $\varepsilon$ at least 7 colors are required. Also, with a sufficiently small $\varepsilon$ the number of colors is exactly 7. The main result of the present paper is that the conjecture is true for the Euclidean plane as well as for any Minkowski plane.