The mixed-motive properad conjecture for framed curves

Let $\mathbfcal M_{g,k}$ denote the logarithmic stack of formal curves with framings, and let $\textsf N^{fr}_{\mathrm{LMM}}$, $\textsf N^{fr}_{\mathrm{MHS}}$, and $\textsf N^{fr}_{\mathrm B}$ denote its conjectural DG-properad in log mixed motives and its Hodge and Betti realisations. For a simply connected compact complex manifold $M$, write $LM$ for its free loop space. The mixed-motive properad conjecture. (i) There exists a DG-properad $\textsf N^{fr}_{\mathrm{LMM}}$ in the category of log mixed motives. (ii) The corresponding Hodge realisation $\textsf N^{fr}_{\mathrm{MHS}}$ acts on the homology of $LM$ compatibly with the mixed Hodge structure on $H_{{\:\raisebox{1pt}{\text{\circle*{1.5}}}}}(LM)$. (iii) The corresponding Betti realisation

$$\textsf N^{fr}_{\mathrm B}:=\{\mathfrak m_{g,k+p}^{an}\}$$

coincides with K. Costello's action. This conjectural package seeks a motivic refinement of the properad of framed curves and its string-topology action; the source attributes the Hodge-compatibility discussion to Hain and the Betti action to Costello.