Aharoni–Kotlar–Ziv matching conjecture for regular simple tripartite 3-graphs

Let $\mathcal{H}$ be an $r$-regular simple $3$-partite $3$-graph with $n$ vertices in each class. Aharoni–Kotlar–Ziv conjecture.

$$\nu(\mathcal{H}) \geq \frac{r - 1}{r}n.$$

This conjecture concerns the matching number of simple regular tripartite $3$-graphs. For $r=n$, it generalizes the open Ryser–Brualdi–Stein problem on Latin transversals, and the conjecture is therefore likely to be difficult in full generality.