Merkulov–Willwacher's properadic correspondence conjecture for framed curves

Let $\Omega(\overline{\textsf N}^{fr})$ be the cobar construction of the properad of framed curves, let $\Omega(\textsf{Frob}^*)$ be the cobar construction of the dual Frobenius properad, let $\textsf{HoLieB}_0^{\diamond}$ be the minimal resolution of $\textsf{LieB}_0^{\diamond}$, and let $\textsf{RGra}_0$ be the properad of genus-zero ribbon graphs. Merkulov–Willwacher's properadic conjecture. There is a morphism, or a roof of morphisms, of properads

$$\Omega(\overline{\textsf N}^{fr})\longrightarrow \textsf{RGra}_0$$

such that the diagram involving the quasi-isomorphism $\Omega(\textsf{Frob}^*)\to\textsf{HoLieB}_0^{\diamond}$, the morphism $\Omega(\textsf{Frob}^*)\to\Omega(\overline{\textsf N}^{fr})$, the morphism $\textsf{HoLieB}_0^{\diamond}\to\textsf{RGra}_0$, and the vertical map to $\textsf{RGra}_0$ commutes. The conjecture proposes a properadic comparison between framed-curve structures and genus-zero ribbon graphs; the paper also states a genus-zero result and discusses constructions intended to establish the conjecture.