Aharoni–Berger–Kotlar–Ziv degree-constrained matching conjecture

Let $\mathcal{H}$ be a simple $3$-partite $3$-graph with vertex classes $A$, $B$ and $C$. Suppose each vertex in $A$ has degree at least $r$, and each vertex in $B \cup C$ has degree at most $r$. Aharoni–Berger–Kotlar–Ziv conjecture.

$$\nu(\mathcal{H}) \geq \frac{r - 1}{r}\left|A\right|.$$

This degree-asymmetric statement extends the regular case and gives a lower bound in terms of the size of one vertex class. For $r=n$, it is among the conjectures that generalize the open Ryser–Brualdi–Stein problem on Latin transversals, so its full generality remains difficult.