Strong antidistinguishability does not imply maximal contextuality

Srikumar et al. introduced a notion of contextuality for finite sets of pure quantum states and proved that maximal contextuality implies strong antidistinguishability. They conjectured the converse, namely that every strongly antidistinguishable non-orthogonal set of pure states is maximally contextual. We give a counterexample in the four-dimensional Hilbert space.