Simultaneous multipartite subhypergraph conjecture

Let $r$ be a positive integer, and let $\mathcal{H}_1,\dots,\mathcal{H}_\ell$ be $r$-uniform hypergraphs on the same vertex set $V$, where $\mathcal{H}_i$ has $m_i$ hyperedges. A hyperedge meets every class of a partition when it has at least one vertex in each class.

Simultaneous multipartite subhypergraph conjecture. There exists a partition of $V$ into $r$ classes $V_1,\dots,V_r$ such that, for every $i=1,\dots,\ell$, at least

$$\frac{r!m_i}{r^r}-o(m_i)$$

hyperedges of $\mathcal{H}_i$ meet each of the classes $V_1,\dots,V_r$.

For a single hypergraph, the leading term follows by randomly partitioning the vertices, and the conjecture asserts that nearly the same proportion can be achieved simultaneously for several hypergraphs. The supplied text does not indicate whether the conjecture is open or resolved.