Asymptotic rainbow matching conjecture for hypergraphs

For each $r\geq2$, let $g(r,n)$ and $g'(r,n)$ be the largest guaranteed rainbow matching sizes for $n$ matchings of size $n$ in, respectively, an $r$-partite $r$-uniform hypergraph and an $r$-uniform hypergraph. Let $h(r,n)$ and $h'(r,n)$ denote the corresponding smallest matching-size parameters guaranteeing a rainbow matching of size $n$.

Asymptotic rainbow matching conjecture. Fix $r\geq2$. For every $x\in\{g(r,n),g'(r,n),h(r,n),h'(r,n)\}$,

$$\lim_{n\to\infty}\frac{x}{n}=1.$$

This conjecture formalises the expected asymptotic sharpness of the principal rainbow matching parameters. The source gives no proof or counterexample, so the assertion remains open.