  Papadima and Suciu proved an inequality between the ranks of the cohomology
groups of the Aomoto complex with finite field coefficients and the twisted
cohomology groups, and conjectured that they are actually equal for certain
cases associated with the Milnor fiber of the arrangement. Recently, an
arrangement (the icosidodecahedral arrangement) with the following two peculiar
properties was found: (i) the strict version of Papadima-Suciu's inequality
holds, and (ii) the first integral homology of the Milnor fiber has a
non-trivial $2$-torsion. In this paper, we investigate the relationship between
these two properties for double covering spaces. We prove that (i) and (ii) are
actually equivalent.
