The $L_{\infty}$-algebra conjecture for the BV double complex

Let $({\bf F}^{\bullet}_{{\bf b}^-},\mathcal{Q})$ be the complex whose degree-two Maurer–Cartan elements encode a Beltrami–Courant differential, vector fields, one-forms, and dilaton shifts, with the auxiliary vector-field and one-form components as described above. The bracket $\{\mathord{\cdot},\mathord{\cdot}\}_-$ is the operation on this complex arising from the BV-double construction.

$L_{\infty}$-algebra conjecture. There exists an $L_{\infty}$ structure on the complex $({\bf F}^{\bullet}_{{\bf b}^-},\mathcal{Q})$, such that its bilinear operation is the symmetrized version of $\{\mathord{\cdot},\mathord{\cdot}\}_-$ and reproduces the Einstein equations and their symmetries as generalized Maurer–Cartan equations and their symmetries.

This conjecture would package the Einstein equations and their symmetry transformations into a generalized Maurer–Cartan framework for the BV double of a Courant algebroid. The supplied text does not state whether the conjecture has been proved or disproved.