Ryser's conjecture for higher uniformities

Prove that every r-partite, r-uniform hypergraph satisfies the Ryser bound \(\tau(\mathcal{H})\leq (r-1)\nu(\mathcal{H})\) for all \(r\geq 4\).

Background

Ryser's conjecture asserts that the vertex-cover number of every r-partite, r-uniform hypergraph is at most $(r-1)$ times its matching number. The paper notes that the conjecture is known for $r=2$ by König's theorem and for $r=3$ by Aharoni, but remains unresolved in every case $r\geq 4$.

Although the paper disproves Lovász's stronger conjecture for $r=3$, it does not resolve Ryser's conjecture for higher uniformities; the latter remains a central open problem in the surrounding discussion.