Configuration-space integral realization conjecture for long embeddings

Let $n$ and $j$ be integers with $n-j\ge 3$. Let $HGC_{n,j}$ be the hairy graph complex, let ${}^*HGC_{n,j}$ denote its dual, and let $DGC_{n,j}$ be the cochain complex equipped with Yoshioka's modified configuration space integral

$$\overline{I}\colon DGC_{n,j}\longrightarrow A_{dR}\bigl(\bEmb_c(\mathbb{R}^j,\mathbb{R}^n)\bigr).$$

For integers $k,g,l$, write $H_l({}^*HGC_{n,j}(k,g))$ for the corresponding homology group, and let $\bEmb_c(\mathbb{R}^j,\mathbb{R}^n)$ denote the space of long embeddings.

Configuration-space integral realization conjecture. There exists a map

$$c\colon H_l({}^*HGC_{n,j}(k,g))\longrightarrow \pi_{(g-1)(j-1)+(n-j-2)k+l}\bigl(\bEmb_c(\mathbb{R}^j,\mathbb{R}^n)\bigr)\otimes\mathbb{R}$$

such that, for every pair $[H]\otimes[\gamma]\in H^{\bullet}(HGC_{n,j})\otimes H_{\bullet}({}^*HGC_{n,j})$, the natural pairing $\langle\overline{I}(H),c(\gamma)\rangle$ coincides with the natural pairing $\langle H,\gamma\rangle$.

This conjecture proposes that the modified configuration space integral detects the relevant rational homotopy classes of long embeddings in codimension at least three through the dual hairy graph complex. The preceding results establish the graph-complex description of rational homotopy in codimension greater than two, while the conjecture asserts compatibility of the geometric integral with the corresponding homology-cohomology pairing.