Balanced counterexamples to Ryser's conjecture at prime orders

For every sufficiently large prime q, we construct a finite intersecting $(q+1)$-partite $(q+1)$-uniform hypergraph with covering number $q+1$ and exactly $q+1$ nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.