Asymptotic uncovered-vertex probability for dense almost-regular hypergraphs

Let $k\geq 3$ be a positive integer and let $0<\delta,\varepsilon<1$ be positive real numbers. There is an $n_0$ such that, for every $n\geq n_0$, if $H$ is an $n$-vertex $k$-graph whose vertex degrees lie between $(1-n^{-\delta})d$ and $(1+n^{-\delta})d$ for some $d>n^{\varepsilon}$, and whose maximum codegree is at most $n^{-\delta}d$, then, for a uniformly random matching $M$ of $H$, writing $\mathcal{P}_H(\overline{v})$ for the probability that $M$ does not cover $v$, the dense almost-regularity conjecture. For every $v\in V(H)$,

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

This is posed in the concluding remarks as the regime in which the disproved regular-linear conjecture may still hold: the paper's counterexamples rely on fixed $d$ and very large constructions, while the claim concerns degrees growing faster than a power of $n$.