The open-closed string analogue of Deligne's conjecture

Let $(B,A,\mathfrak l,\mathfrak q)$ be an open-closed homotopy algebra, and let $C^{\bullet,\bullet}(B;A,A)$ denote its open-closed cochain space.

Open-closed Deligne conjecture. For any open-closed homotopy algebra $(\mathfrak l,\mathfrak q)$, there is a $G_\infty$ algebra structure on the space $C^{\bullet,\bullet}(B; A, A)$.

This conjectures an open-closed string analogue of Deligne's Hochschild cohomological conjecture, which asserts that Hochschild cochains carry a homotopy Gerstenhaber structure. The corresponding closed-string result is known through resolutions of Deligne's conjecture and the formality of the little-disks operad; an affirmative resolution of the open-closed version is left for future work.