Asymmetric degree conjecture for 3-partite hypergraph matchings

Let $H$ be a simple $3$-partite hypergraph with sides $A$, $B$, and $C$. Write $\delta(A)$ for the minimum degree of a vertex in $A$, $\Delta(B\cup C)$ for the maximum degree of a vertex in $B\cup C$, and let $\nu(H)$ be the maximum matching size. Asymmetric degree conjecture. The following assertions should hold:

1. If $d=\delta(A)\ge \Delta(B\cup C)$, then

$$\nu(H)\ge \frac{d-1}{d}|A|.$$

2. If

$$\delta(A)\ge \max\bigl(\Delta(B\cup C),2|A|-1\bigr),$$

then

$$\nu(H)=|A|.$$

This formulation is intended to capture the essence of the regular conjecture in an asymmetric setting; the paper gives only partial results.