Enhanced derived-Picard-group conjecture for closed-surface Fukaya categories

The conjecture seeks a complete description of the derived Picard group of a closed-surface Fukaya category, including recovery of the surface genus. The proposed description is

$$(H^{1}(\Sigma_g;\Lambda^{\times})\rtimes\pi_{0}\mathrm{Diff}^{+}(\Sigma_g))\times\mathbb{Z}/2\mathbb{Z}.$$

Known results

• A June 2020 paper proves that, for $g\ge 2$, spherical objects with nonzero Chern character come from simple closed curves with rank-one local systems, and gives a surjection $\mathrm{Auteq}(D^{\pi}\mathcal{F}(\Sigma_g))\to\Gamma(\Sigma_g)$.
• The same paper explicitly leaves the full autoequivalence-group description as a conjecture, proposing $H^{1}(\Sigma_g;\Lambda^{*})\rtimes\Gamma(\Sigma_g)$.

September 2026 preprint

Dongjian Wu and Nantao Zhang report the claimed closed-surface formula and genus recovery in a September 2026 preprint. No independent mathematical assessment was found, so this does not establish a solution.

Current status (as of September 2026): The earlier partial results are established, but the enhanced conjecture remains open because the new resolution claim is unverified.