We prove that $χ(G) \le \lceil (Δ+1)/2\rceil+1$ for any triangle-free graph $G$ of maximum degree $Δ$ provided $Δ\ge 524$. This gives tangible progress towards an old problem of Vizing, in a form cast by Reed. We use a method of Hurley and Pirot, which in turn relies on a new counting argument of the second author.