Yuster's triangle-packing conjecture for random graphs

Let $G(n,p)$ be the binomial random graph, and let a $K_3$-packing be a collection of pairwise edge-disjoint triangles. Its leave consists of the edges not covered by the packing.

Yuster's triangle-packing conjecture. If

$$p\geq (1+\varepsilon)\sqrt{\frac{\log n}{n}},$$

then asymptotically almost surely $G(n,p)$ has a $K_3$-packing containing all but at most $3n$ edges.

This conjecture concerns the threshold for packing triangles with a small leave. The surrounding discussion reports little progress on Yuster's conjectures and presents the paper's results as improvements to the known bounds, so the conjecture remains open in the supplied source.