Wegner's conjecture on good covers

Let a good cover in $\mathbb{R}^d$ be a collection of open sets such that every subcollection has intersection either empty or homeomorphic to an open $d$-ball. The nerve of a collection is the simplicial complex whose vertices are the sets and whose faces are the subcollections with nonempty intersection. A finite simplicial complex is topologically $d$-representable if it is isomorphic to the nerve of a good cover in $\mathbb{R}^d$. A simplicial complex is $d$-collapsible if it can be reduced to the empty complex by repeatedly removing a face of dimension at most $d-1$ contained in a unique maximal face. Wegner's conjecture. Every topologically $d$-representable simplicial complex is $d$-collapsible. This conjecture proposes a topological extension of Wegner's theorem that every convexly $d$-representable simplicial complex is $d$-collapsible; the paper gives a counterexample, so the conjecture is refuted.