Structured-cover conjecture for intersecting multipartite hypergraphs

Let $H$ be an intersecting $r$-partite hypergraph with sides $V_1,\ldots,V_r$. A cover is a set of vertices meeting every edge. The structured-cover conjecture. There exists either a side of size at most $r-1$, or an edge $e\in H$ and a vertex $x\in e$ such that $e\setminus\{x\}$ is a cover.

This would be a stronger form of Ryser's conjecture for intersecting hypergraphs. A fractional version is proved in the paper, but the integral assertion remains open.