Vassiliev's homotopy-splitting conjecture for the resolved discriminant

Let $\sigma$ be the simplicial resolution of the discriminant $\Sigma$ of the space of long knots, and let

$$\emptyset=\sigma_0\subset\sigma_1\subset\sigma_2\subset\dots$$

be its natural filtration. Write $\bar\sigma_i$ and $\bar\sigma$ for the one-point compactifications of the filtration terms and of the full resolution, respectively. Vassiliev's homotopy-splitting conjecture. The filtration homotopically splits, meaning that $\bar\sigma$ is homotopy equivalent to

$$\bigvee_{i=1}^{+\infty}(\bar\sigma_i/\bar\sigma_{i-1}).$$

The assertion would decompose the resolved discriminant into the successive filtration quotients and is closely related to simplifying the spectral-sequence computation of long-knot homology.