On profinite groups with polynomially bounded Mobius numbers

A conjecture of A. Mann asks if every finitely generated profinite group that is PFG has PBMN. Here a group G is said to be positively finitely generated (PFG) if the probability P(G,k) that k randomly chosen elements generate G topologically is positive, for some k. And G is said to have polynomially bounded Möbius numbers (PBMN) if the absolute value of the Möbius function μ(H,G) is bounded by a polynomial function in the index in G of the subgroup H, and the number b(n,G) of subgroups H of index n with μ(H,G) not equal to zero grows at most polynomially in n. We obtain a quantitative reduction theorem which requires to deal only with almost simple groups.