The terminal leave in random greedy hypergraph matching

Let $G$ be the terminal graph of the random triangle-removal process started from $K_n$. We prove that $e(G)\mathrm{/}{n}^{3\mathrm{/}2}\to 1\mathrm{/}(2\sqrt{2})$ in $L^2$. More generally, we obtain the analogous result for any fixed strictly $k$-balanced $k$-graph. Our main tool is a more general theorem on random greedy matchings in nearly regular uniform hypergraphs with few short cycles under a stability assumption on the survival equations.