Gowers norms of multiplicative functions in progressions on average

Let μ be the Möbius function and let k ≥ 1. We prove that the Gowers U^k-norm of μ restricted to progressions {n ≤ X : n ≡ a_q (mod q)} is o(1) on average over q ≤ X^1/2-σ for any σ > 0, where a_q (mod q) is an arbitrary residue class with (a_q, q) = 1. This generalizes the Bombieri-Vinogradov inequality for μ, which corresponds to the special case k = 1.