Closed-string polyvector field conjecture for symplectic cluster manifolds

Let $M_\mathcal{R}$ be the symplectic cluster manifold, let $\mathcal{Y}_\mathcal{R}$ be its mirror, and suppose that all node multiplicities in $\mathcal{R}$ are $1$. Write $SH_{M_\mathcal{R}}^k(M_\mathcal{R},\Lambda)$ for symplectic cohomology and $H^p(\mathcal{Y}_\mathcal{R},\Lambda^qT\mathcal{Y}_\mathcal{R})$ for Čech cohomology of polyvector fields with respect to the defining affinoid cover. Using $\Omega_\mathcal{R}$, equip the polyvector fields with the differential transported from the de Rham differential. Closed-string polyvector field conjecture. There exists a ring isomorphism

$$SH_{M_\mathcal{R}}^k(M_\mathcal{R},\Lambda)\to\bigoplus_{p+q=k}H^p(\mathcal{Y}_\mathcal{R},\Lambda^qT\mathcal{Y}_\mathcal{R}),$$

where the right-hand side is the Čech cohomology with respect to the defining affinoid cover; moreover, the transported differential on polyvector fields is compatible with the BV operator. The conjecture is motivated by a proposed degeneration of the relevant spectral sequence and by local closed-open and Hochschild–Kostant–Rosenberg comparisons; its general validity remains open.