Conjecture on lifting the Green witness to an $L_{\infty}$ algebra

Let $(F,Q,(-,-),W)$ be a free BV field theory, and suppose its Green's witness $W$ is obtained by truncating an $L_{\infty}$ algebra to its underlying cochain complex. Green-witness lifting conjecture. The Green's witness $W$ may be lifted to the level of that $L_{\infty}$ algebra, perhaps only up to homotopy. This would extend the preceding linearized construction to perturbative field theories modelled by $L_{\infty}$ algebras or differential graded Lie algebras.