\v{S}olt\'es' hypergraphs

More than $30$ years ago, \v{S}olt\'es 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 \v{S}olt\'es graphs. We extend the definition of \v{S}olt\'es' graphs to \v{S}olt\'es' hypergraphs, determine all orders for which a \v{S}olt\'es' hypergraph exists, observe infinitely many uniform \v{S}olt\'es' hypergraphs, and find the \v{S}olt\'es' hypergraph with minimum size (spoiler: it is not $C_{11}$).