Weaker deletion conjecture implying Ryser's conjecture

Prove or disprove the proposed weakening that every r-partite, r-uniform hypergraph contains, for some \(k\) with \(1\leq k\leq r-1\), \(k(r-1)\) vertices whose deletion reduces the matching number by at least \(k\).

Background

Lovász's original conjecture is false, as shown by the paper's Biggs–Smith line-hypergraph counterexample. The authors therefore propose a weaker statement that permits deleting a multiple of $r-1$ vertices and requires a corresponding matching-number reduction of at least the multiplier.

The proposed weakening is intended to remain strong enough to imply Ryser's conjecture. The authors verify it for the line hypergraph of the Biggs–Smith graph but leave its validity in general unresolved.