Functorial Goldman–Turaev formality conjecture for framed surfaces

Let $\boldsymbol{\widehat{\mathbf{PaB}}}^f$ be the prounipotent completion of the operad of parenthesized framed braids, let $\boldsymbol{\widehat{\mathbf{PaB}}}^f_g$ be the corresponding operad module for a compact oriented surface of genus $g$, and let $F_1$ denote a full $360^\circ$ rotation of the framing. For $n\geq 0$, let $f$ be a framing on the surface $\Sigma_{g,n+1}$, and let $\overline{K}_n$, $K_n$, $M$, $U\overline{K}_{n+2}$, and $\mathrm{OpR}^\Delta_f(n+2)$ be the objects and spaces defined in the paper. Functorial Goldman–Turaev conjecture. There exists a section

$$\gamma^f:\overline{K}_{n+2}\longrightarrow K_{n+2}$$

and an isomorphism of $\overline{K}_n\oplus\overline{K}_n$-modules

$$\varphi:M\longrightarrow U\overline{K}_{n+2}$$

such that

$$((\widehat{\mathbf{PaB}}^f,\widehat{\mathbf{PaB}}^f_g,F_1),\gamma^f,\varphi)\in\mathrm{OpR}^\Delta_f(n+2),$$

and such that the map $\Lambda_{n+2}$ sends this structure to

$$(|\widehat{\mathbb{K}\pi}|,[-,-],\delta^f),$$

the Goldman–Turaev bialgebra on $\Sigma_{g,n+1}$ associated with $f$. The conjecture proposes a functorial algebraic realization of the Goldman–Turaev bialgebra from the completed parenthesized framed braid operad and its surface module; the supplied text gives no resolution evidence, so its status remains open.