A note on multiplicative functions on progressions to large moduli

Let f : N → C be a bounded multiplicative function. Let a be a fixed nonzero integer (say a = 1). Then f is well-distributed on the progression n ≡ a(mod q) ⊂ {1,...,X}, for almost all primes q ∈ [Q; 2Q], for Q as large as X1/2 + 1/78-o(1).