Simple 3-partite hypergraph generalization of Drisko's theorem

Let $H$ be a simple $3$-partite hypergraph with sides $A$, $B$, and $C$, where simplicity means that no edge is repeated. Let $\nu(H)$ denote the maximum matching size. Generalized Drisko conjecture. If

$$|A|=2n-1,$$ $$\deg(a)\ge n\quad\text{for every }a\in A,$$ $$\deg(v)\le 2n-1\quad\text{for every }v\in B\cup C,$$

then

$$\nu(H)\ge n.$$

This is proposed as a simple-hypergraph generalization of the sharp $2n-1$ rainbow matching theorem; its status is not resolved in the supplied text.