A rate for the average vacancy of uniformly random matchings in linear hypergraphs

Let $M$ be chosen uniformly from all matchings of a finite linear $k$-uniform hypergraph $H$, and let $\overline{q}(H)$ be the average probability that a vertex is left uncovered. If $H$ has maximum degree $D$ and normalized average degree $β=k|E(H)|/(|V(H)|D)$, then, for every fixed $k\geq 2$ and u