Bosek's arithmetic graph chromatic conjecture

For a positive integer $k$, define the arithmetic proximity of positive integers $a,b$ by $\max\{a/d,b/d\}$, where $d=\gcd(a,b)$. The arithmetic graph $B_k$ has vertex set $\mathbb N$, with $a$ adjacent to $b$ exactly when their arithmetic proximity is at most $k$. Bosek's arithmetic graph chromatic conjecture. Every arithmetic graph $B_k$ satisfies $\chi(B_k)=k$. This connects ironic decorations with arithmetic structure and remains open.