Vassiliev's first-term stabilization conjecture for the long-knot spectral sequence

Let $l:{\mathbb R}^1\hookrightarrow{\mathbb R}^d$ be a fixed non-trivial linear map, let ${\mathcal K}$ be the affine space of smooth maps ${\mathbb R}^1\to{\mathbb R}^d$ agreeing with $l$ outside a compact set, and let $\Sigma\subset{\mathcal K}$ be the discriminant of non-injective or singular maps. Let $\sigma$ be the simplicial resolution of $\Sigma$, with filtration

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

Vassiliev's stabilization conjecture. The spectral sequence associated with this filtration and computing the Borel–Moore homology groups of the resolution $\sigma$ stabilizes over $\mathbb Q$ in the first term. This concerns the spectral sequence used to compute the homology of the resolved discriminant, and the conjecture gives a particularly simple form of its convergence behavior.