Kahn's matching-uncovered probability conjecture for regular linear hypergraphs

Let $H$ be a $d$-regular linear $k$-graph, and let $M$ be a matching chosen uniformly from the set of all matchings of $H$. For a vertex $v$, write $\mathcal{P}_H(\overline{v})$ for the probability that $M$ does not cover $v$. Kahn's conjecture. For every $v\in V(H)$,

$$\mathcal{P}_H(\overline{v})=(1+o_d(1))d^{-1/k}.$$

Kahn proposed this as an asymptotic description of the average size of random matchings in regular linear hypergraphs. It was proved for graphs ($k=2$) by Kahn and Kim, but the paper disproves it for every $k\geq 3$; thus the conjecture is refuted in those cases.