Ozsváth–Szabó's spectral sequence differential conjecture for torsion \a0Spin^c three-manifolds

Let $Y$ be a closed, oriented three-manifold equipped with a torsion ${\mathrm{Spin}}^c$ structure $\mathfrak t_0$. The spectral sequence for $HF^\infty(Y,\mathfrak t_0)$ has

$$E_2=\mathbb{Z}[U,U^{-1}]\otimes_{\mathbb{Z}}\Lambda^*H^1(Y;\mathbb{Z}).$$

Spectral sequence differential conjecture. The spectral sequence collapses after the $E_3$ stage, and its differential

$$d_3\colon \Lambda^{i}H^1(Y;\mathbb{Z})\otimes_{\mathbb{Z}} U^j \longrightarrow \Lambda^{i-3} H^1(Y;\mathbb{Z})\otimes_{\mathbb{Z}} U^{j-1}$$

is given by

$$d_3(\phi_1\wedge\ldots\wedge\phi_i) =\frac{1}{3!\cdot(i-3)!}\sum_{\sigma\in{\mathfrak S}_i} (-1)^\sigma \langle \phi_{\sigma(1)}\cup\phi_{\sigma(2)}\cup \phi_{\sigma(3)},[Y]\rangle\, \phi_{\sigma(4)}\wedge\ldots\wedge\phi_{\sigma(i)}.$$

Here ${\mathfrak S}_i$ is the permutation group on $i$ letters, and $(-1)^\sigma$ is the sign of the permutation. This conjecture predicts that the triple-cup-product pairing determines the only potentially nonzero higher differential in the $HF^\infty$ spectral sequence. The preceding calculations for $T^3$ and $S^1\times\Sigma_2$ are consistent with it, but the general collapse and differential formula are not established in the supplied source.