Non-acyclic spaces of line transversals

Cheong, Goaoc and Holmsen conjectured that every connected component of the space of line transversals to a family of pairwise disjoint open convex sets in ${\mathbb{R}}^d$ is acyclic. We disprove this conjecture by showing that the homology may be nontrivial in any fixed dimension provided that $d$ is large enough. More precisely, we show that for every $n\ge 1$ there is a finite family of pairwise disjoint open convex sets in ${\mathbb{R}}^{3n}$ such that the $(n-1)$st homology (over an arbitrary ring) of the space of line transversals to this family is nonzero.