The cyclic-polytope conjecture for unique rainbow simplices

Let $K$ be a triangulation of the $d$-simplex $\Delta^d$ associated with a $d$-dimensional cyclic polytope, meaning that it is obtained from the boundary of the polytope by removing a facet and projecting the remaining boundary into the convex hull of the vertex set of that facet. Problem~ asks whether there is a Sperner labelling of $K$ with a unique rainbow $d$-simplex. Cyclic-polytope conjecture. Problem~ has an affirmative answer if $K$ is a triangulation of $\Delta^d$ associated with a $d$-dimensional cyclic polytope. The question concerns whether triangulations of this special, well-understood type can have the unique-rainbow-simplex property in higher dimensions; the supplied text gives no resolution.