The sharp terminal leave in random triangle removal

Starting from the complete graph on n vertices, repeatedly remove the three edges of a uniformly chosen remaining triangle. We prove that the number of edges left at termination, divided by n${}^{3/2}$, converges in L${}^2$ to $1/(2\sqrt{2})$. This proves the triangle case of the sharp-constant conjecture of Joos and Kühn. In particular, the same limit holds in probability and for the normalized expectation.