Šoltés' hypergraphs

More than $30$ years ago, Šoltés observed that the total distance of the graph $C_{11}$ does not change by deleting a vertex, and wondered about the existence of other such graphs, called Šoltés graphs. We extend the definition of Šoltés’ graphs to Šoltés’ hypergraphs, determine all orders for which a Šoltés’ hypergraph exists, observe infinitely many uniform Šoltés’ hypergraphs, and find the Šoltés’ hypergraph with minimum size (spoiler: it is not $C_{11}$).