Criticality in Sperner's Lemma

We answer a question posed by T. Gallai in 1969 concerning criticality in Sperner's lemma, listed as Problem 9.14 in the collection of Jensen and Toft [Graph coloring problems, John Wiley & Sons, Inc., New York, 1995]. Sperner's lemma states that if a labelling of the vertices of a triangulation of the $d$-simplex $\Delta^d$ with labels $1, 2, \ldots, d+1$ has the property that (i) each vertex of $\Delta^d$ receives a distinct label, and (ii) any vertex lying in a face of $\Delta^d$ has the same label as one of the vertices of that face, then there exists a rainbow facet (a facet whose vertices have pairwise distinct labels). For $d\leq 2$, it is not difficult to show that for every facet $\sigma$, there exists a labelling with the above properties where $\sigma$ is the unique rainbow facet. For every $d\geq 3$, however, we construct an infinite family of examples where this is not the case, which implies the answer to Gallai's question as a corollary. The construction is based on the properties of a $4$-polytope which had been used earlier to disprove a claim of T. S. Motzkin on neighbourly polytopes.