On a cardinal invariant related to the Haar measure problem

In [6], given a metrizable profinite group G, a cardinal invariant of the continuum fm(G) was introduced, and a positive solution to the Haar Measure Problem for G was given under the assumption that non(N) ≤ fm(G). We prove here that it is consistent with ZFC that there is a metrizable profinite group G_* such that non(N) > fm(G_*), thus demonstrating that the strategy of [6] does not suffice for a general solution to the Haar Measure Problem.