The cubical knot complex is homotopy equivalent to the Vassiliev simplicial complex

Let $CK^*$ be the space obtained from $CK_b(S^3)$ by attaching an $(n+1)$-cell for every knot $K_{(n-2,1)}$, with $n\geq 2$, whose boundary is given by resolving the double points and the triple point as specified. Cubical–Vassiliev homotopy conjecture. The space

$$\bigcup_n \Sigma^n(CK^*)$$

is homotopy equivalent to the Vassiliev simplicial complex. This proposes that the cubical complex enlarged by cells for triple-point singularities recovers the homotopy type of the Vassiliev complex; the supplied text gives no evidence that the claim has been resolved.