Ishibashi–Sugawara–Yoshinaga torsion conjecture for arrangement double covers

Let $\mathcal{A}$ be a complex hyperplane arrangement in $\mathbb{C}^n$, and let $M(\mathcal{A})$ be its complement. For a nonzero $\omega\in H^1(M(\mathcal{A}),\mathbb{Z}_2)$, write $M(\mathcal{A})^\omega$ for the associated double covering and $\mathcal{L}_\omega$ for the corresponding rank-one local system. Ishibashi–Sugawara–Yoshinaga's conjecture. The integral homology $H_1(M(\mathcal{A})^\omega,\mathbb{Z})$ has nontrivial $2$-torsion if and only if the local-system homology $H_1(M(\mathcal{A}),\mathcal{L}_\omega)$ has nontrivial $4$-torsion. The conjecture is stated for arrangement complements; the cited corollary gives a positive answer because such complements are homotopy equivalent to minimal CW complexes.