Lovász's matching-reduction conjecture for r-partite hypergraphs

Let $H$ be an $r$-partite hypergraph containing at least one edge, and let $\nu(H)$ denote its matching number, the maximum number of pairwise disjoint edges. For a set $S$ of vertices, write $H-S$ for the hypergraph obtained by deleting $S$ and all incident edges. Lovász's conjecture. There exists a set $S$ of $r-1$ vertices such that

$$\nu(H-S)<\nu(H).$$

This conjecture was proposed by Lovász in 1975 as a possible iterative strategy for proving Ryser's conjecture. It has been disproved for $r=3$ and $r=4$, so the database status is refuted.