Yuster's triangle-decomposition conjecture for random regular graphs

Let $G_{n,d}$ be the uniformly random $d$-regular graph on $n$ vertices. A graph is $K_3$-divisible when its degrees are even and its number of edges is divisible by $3$; for a $d$-regular graph this requires $d$ even and $3\mid dn$.

Yuster's triangle-decomposition conjecture. If $d\gg\sqrt{n\log n}$ and $d$ is even, then asymptotically almost surely $G_{n,d}$ has a $K_3$-decomposition provided $3\mid dn$.

This is the random-regular-graph counterpart of Yuster's triangle-packing conjecture. The supplied source does not report a resolution, so it remains open.