Munson–Volić collapse conjecture for the long-link spectral sequence

Let $\mathcal{K}^{m\bullet}_d$ be the cosimplicial space of long links of $m$ strands in $\mathbb{R}^d$ constructed by Munson and Volić, and let its homology Bousfield–Kan spectral sequence converge to $H_*(\overline{\mathcal{L}}_m^d;\mathbb{Q})$. Munson–Volić's conjecture. This spectral sequence collapses at the $E^2$ page rationally for $d\geq 4$. The paper states that its main result proves this conjecture, so the conjecture is solved.