The critical-eigenspace conjecture for completed symplectic cohomology

Let $M$ be the symplectic manifold under consideration and let $\Lambda$ be its Novikov coefficient ring. Let $(\overline{SC}_\Lambda,\partial)$ be the completion of the symplectic cochain complex with respect to the $\widetilde{\mathcal{Q}}$-filtration. For each divisor $D_i$ with parameter $\lambda_i$, let $QH^*(M;\Lambda)_i$ be the generalized zero-eigenspace of quantum multiplication by $\operatorname{PD}(D_i)$, namely the subspace of $\alpha\in QH^*(M;\Lambda)$ such that

$$\operatorname{PD}(D_i)^{\star k}\star\alpha=0$$

for some $k$, and define

$$QH^*(M;\Lambda)_{\mathrm{crit}}=\bigcap_{i:\lambda_i>2}QH^*(M;\Lambda)_i.$$

The critical-eigenspace conjecture. There is an isomorphism

$$H^*(\overline{SC}_\Lambda,\partial)\cong QH^*(M;\Lambda)_{\mathrm{crit}}.$$

Furthermore, the resulting spectral sequence converges to $QH^*(M;\Lambda)_{\mathrm{crit}}$.

This conjecture describes the cohomology obtained after completing the filtered symplectic complex when the grading hypothesis needed for convergence of the uncompleted complex is absent. It is motivated by suggestions of Pomerleano and Seidel; the asserted identification and convergence remain open in the supplied source.