Ozsváth–Szabó's triple-cup-product conjecture for the Heegaard Floer spectral sequence

Let $Y$ be a closed, oriented three-manifold and let $\mathfrak{s}$ be a torsion $mathop{\rm Spin}^c$ structure. The spectral sequence with

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

converges to $HF^\infty(Y,\mathfrak{s})$. For $i,j\in\mathbb{Z}$, the differential has the form

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

Ozsváth–Szabó's conjecture. The differential $d_3$ is given by

$$\phi^1\wedge\cdots\wedge\phi^i\longmapsto\frac{1}{3!(i-3)!}\sum_{\sigma\in S_i}(-1)^{|\sigma|}\left\langle\phi^{\sigma(1)}\smile\phi^{\sigma(2)}\smile\phi^{\sigma(3)},[Y]\right\rangle\,\phi^{\sigma(4)}\wedge\cdots\wedge\phi^{\sigma(i)},$$

and all higher differentials vanish. Thus the spectral-sequence differentials are completely determined by the integral triple cup product form of $Y$.