Functoriality of the L-infinity structure under exact Lagrangian cobordisms

Let $\Lambda^-$ and $\Lambda^+$ be pointed Legendrian links, and suppose that there is an exact Lagrangian cobordism from $\Lambda^-$ to $\Lambda^+$. Let $((\mathcal{A}^\pm)^{\operatorname{comm}},\{\ell_k^\pm\})$ denote the associated homotopy Poisson algebras, and let $(\mathcal{A}^+)^{\operatorname{comm}}\to(\mathcal{A}^-)^{\operatorname{comm}}$ be the usual cobordism map of Chekanov--Eliashberg DGAs.

Functoriality conjecture. There is an $L_\infty$ morphism from the $L_\infty$ algebra $((\mathcal{A}^+)^{\operatorname{comm}},\{\ell_k^+\})$ of $\Lambda^+$ to the $L_\infty$ algebra $((\mathcal{A}^-)^{\operatorname{comm}},\{\ell_k^-\})$ of $\Lambda^-$ extending the usual cobordism map of Chekanov--Eliashberg DGAs $(\mathcal{A}^+)^{\operatorname{comm}}\to(\mathcal{A}^-)^{\operatorname{comm}}$.

This conjecture proposes that the structure is functorial for exact Lagrangian cobordisms, extending the known functoriality of the Chekanov--Eliashberg DGA.