Canonical $L_{\infty}$ morphism for stable generalized complex structures

Let $\Omega^{\bullet}_{\mathbb J,H}$ denote the total complex of the double complex in~, equipped with the zero bracket, and let the total complex of~ carry its $L_{\infty}$ algebra structure. The map on Maurer–Cartan elements is defined by

(\beta,\eta)\longmapsto \bigl(\widetilde\mathrm{a}^{*}(\beta(1-\pi\beta)^{-1}),\eta\bigr).

Canonical $L_{\infty}$ morphism conjecture. There is a canonical $L_{\infty}$ morphism from $\Omega^{\bullet}_{\mathbb J,H}$ to the total complex of~, extending the given quasi-isomorphism, whose induced map on Maurer–Cartan elements is the map above.

This conjecture would connect the formal deformation problem for the pair consisting of a complex log symplectic structure and a real closed 3-form with the deformation theory of stable generalized complex structures. The source suggests that it should be tractable by combining existing results, but no resolution is supplied here.