The Chen-Lih-Wu Conjecture states that each connected graph with maximum degree $Δ\geq 3$ that is not the complete graph $K_{Δ+1}$ or the complete bipartite graph $K_{Δ,Δ}$ admits an equitable coloring with $Δ$ colors. For planar graphs, the conjecture has been confirmed for $Δ\geq 13$ by Yap and Zhang and for $9\leq Δ\leq 12$ by Nakprasit. In this paper, we present a proof that confirms the conjecture for graphs embeddable into a surface with non-negative Euler characteristic with maximum degree $Δ\geq 9$ and for planar graphs with maximum degree $Δ\geq 8$.