Quasi-isomorphism conjecture for link and braid diagram complexes

For $n\geq 3$, let $\mathcal{L}_{m}^{n}$ be the space of long links of $m$ components in $\mathbb{R}^{n}$, and let $\mathcal{B}_{m}^{n}$ be the space of flattened pure braids. For each $o>0$ and $d\geq 0$, let $\mathcal{LD}^{o,d}$ and $\mathcal{BD}^{o,d}$ be the corresponding diagram complexes, with cochain maps given by configuration space integrals:

$$\mathcal{LD}^{o,*} \longrightarrow \Omega^{(n-3)o+*}(\mathcal{L}_{m}^{n}),\qquad \mathcal{BD}^{o,*} \longrightarrow \Omega^{(n-3)o+*}(\mathcal{B}_{m}^{n}).$$

Quasi-isomorphism conjecture. The cochain maps from Theorem 1 are quasi-isomorphisms for $n>3$. This conjecture asserts that the diagram complexes calculate the cohomology of spaces of long links and flattened pure braids in dimensions greater than three; the supplied text gives no resolution status.