Cheong–Goaoc–Holmsen acyclicity conjecture for spaces of line transversals

Let $d$ be a positive integer, and let ${\mathcal F}$ be a family of at least two pairwise disjoint open convex sets in ${\mathbb R}^d$. A line in ${\mathbb R}^d$ is transversal to ${\mathcal F}$ if it meets every member of ${\mathcal F}$; the space of line transversals is the topological space formed by all such lines. Cheong–Goaoc–Holmsen conjecture. Every connected component of the space of line transversals to ${\mathcal F}$ is acyclic. Cheong, Goaoc and Holmsen proved the assertion in dimension $3$, while the paper disproves its higher-dimensional generalization by constructing, for every $n\geq 1$, a finite family in ${\mathbb R}^{3n}$ whose space of line transversals has nonzero $(n-1)$st homology over an arbitrary ring.