Lu–Ma's non-trivial matching conjecture for partite hypergraphs

For integers $r\ge 3$ and $n_1\ge\dots\ge n_r\ge 2$, let $X_1,\dots,X_r$ be disjoint sets with $|X_i|=n_i$. Define $m_0(s,n_1,\dots,n_r)$ as the maximum size of a family $\mathcal{F}\subseteq X_1\times\dots\times X_r$ satisfying $\nu(\mathcal{F})\le s<\tau(\mathcal{F})$; in the symmetric case write $m_0(s,n;r)=m_0(s,\underbrace{n,\dots,n}_{r})$. Lu–Ma's conjecture. Let $n$ be sufficiently large, $r\ge4$, and $s<n$. Then

$$m_0(s,n,\dots,n)=sn^{r-1}-(n-1)^{r-1}+n-s.$$

The conjecture was confirmed in the paper for $r\ge3$ and $s=1$, for fixed $r\ge3$ and $s\ge2$ when $n$ is sufficiently large, and previously for $r=3$ under a separate sufficiently-large-$n$ condition; the general statement as formulated here remains open.