Combinatorial dga model conjecture for surface categories

Let $(S,n,{\bf a})$ be a parametrized surface and let $R(S,n,{\bf a};k)$ be the dga generated by diagrams of $k$ strands between the corresponding collections of arcs. Let $\mu_m$ denote the higher $A_{\infty}$-operations for the associated surface category. Surface dga model conjecture. The operations $\mu_m$ are trivial for $m\geq 3$, the resulting structure is isomorphic to $R(S,n,{\bf a};k)$, and the quasi-isomorphism class of $R(S,n,{\bf a};k)$ is independent of the choice of parametrization ${\bf a}$ of $S$. These conjectures would provide a combinatorial description of the partially wrapped $A_{\infty}$-algebras from the geometric construction; the paper presents them as conjectural and gives partial justification.