In the well known Cohen - Lenstra paper published in 1983, the authors present an experimentally well-supported conjecture on computing certain asymptotics of class groups of real abelian and complex quadratic $\mathbb{Q}$ field extensions. Few of the resulting sub conjectures have already been proven by Davenport-Heilbron, Bhargava, et al and there is ongoing research on these sub conjectures.
But I would like to know if there have been or ongoing efforts to prove their main conjecture that they state under "Section 8.1 Fundamental assumptions". At least for the case of complex quadratic extensions. Are there any references to such attempts?
