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