Before revision:  In 2019, Correa et al. showed that when the distributions differ, it is no longer possible to achieve this bound: they prove an upper bound of \sqrt{3 - 1 < 0.74 on the best approximation ratio.
After revision:  In 2019, Correa et al. showed that when the distributions differ, it is no longer possible to achieve this bound: they prove an upper bound of \sqrt{3 -