The list version of the Borodin-Kostochka conjecture for graphs with large maximum degree

Brooks' Theorem states that the chromatic number χ(G) of a graph G is at most its maximum degree Δ(G) when Δ(G)≥3 and its clique number ω(G) is at most Δ(G). Vizing strengthened this by proving that the list chromatic number χ_l(G) is also at most Δ(G) for these graphs. Borodin and Kostochka conjectured that χ(G)<Δ(G) for all graphs G with Δ(G)≥9 and ω(G)<Δ(G). Reed proved their conjecture for graphs G with Δ(G) sufficiently large. Here we prove the analogous result for list coloring: χ_l(G)<Δ(G) for all graphs G with Δ(G) sufficiently large and ω(G)<Δ(G).