Counterexamples in multimarginal optimal transport with Coulomb cost and spherically symmetric data

We disprove a conjecture in Density Functional Theory, relative to multimarginal optimal transport maps with Coulomb cost. In the case of spherically symmetric data, which model for instance Lithium and Beryllium atoms, we show that some special maps, introduced by Seidl, Gori-Giorgi and Savin are not always optimal in the corresponding transport problem. We also provide examples of maps satisfying optimality conditions for special classes of data.