  It was believed that modular data are enough to distinguish different modular
categories (and topological orders in 2+1-dimensions). Then counterexamples to
this conjecture were found by Mignard and Schauenburg in 2017. In this work, we
show that the simplest counterexamples can be distinguished by studying the
representations of mapping class groups of a punctured torus or a genus-2
closed manifold.
