Generalized structured-cover conjecture for multipartite hypergraphs

Let $H$ be an $r$-partite hypergraph with matching number $u(H)=k$. A set is contained in a side or in an edge if it is a \subset of one of the sides or of one edge, respectively. Generalized structured-cover conjecture. There exist sets $S_1,\ldots,S_k$, each of size at most $r-1$ and contained in a side or in an edge, such that

$$\bigcup_{i\leqslant k}S_i$$

is a cover of $H$.

This would generalize the preceding structured-cover assertion and imply a strengthened form of Ryser's conjecture. The paper presents it as open.