Seidel–Dostoglou decomposition conjecture for Floer cohomology of

Let $\Sigma$ be a closed oriented surface of genus $g$, let $s^i\Sigma$ denote its $i$-th symmetric product, and let $HF_g^*$ be the Floer cohomology ring associated with $\Sigma\times S^1$. Assume the eigenspace decomposition in equation~ is defined, with eigenvalues indexed by triples of the form $(\pm 4(g-1-i)\sqrt{-1}^{g-i},(-1)^{g-1-i}8,0)$. Seidel–Dostoglou decomposition conjecture. There is a $\mathbb{Z}/2\mathbb{Z}$-graded, $\operatorname{Diff}(\Sigma)$-equivariant isomorphism

$$HF_g^* \cong H^*(s^0\Sigma)\oplus H^*(s^1\Sigma)\oplus\cdots\oplus H^*(s^{g-2}\Sigma)\oplus H^*(s^{g-1}\Sigma)\oplus H^*(s^{g-2}\Sigma)\oplus\cdots\oplus H^*(s^0\Sigma),$$

where $H^*(s^i\Sigma)$ is isomorphic to the eigenspace with eigenvalues $(\pm 4(g-1-i)\sqrt{-1}^{g-i},(-1)^{g-1-i}8,0)$. The isomorphism respects only the $\mathbb{Z}/2\mathbb{Z}$-grading. This conjecture predicts that the Floer cohomology is organized by the cohomology of symmetric products of the surface and is compatible with the mapping class group action, but the supplied text gives no evidence that the conjecture has been proved or disproved.