The latest results are not very recent. Most of them are mentioned in the paper

> MR1162188 Carleson, Lennart; Jones, Peter W. On coefficient problems for univalent functions and conformal dimension. Duke Math. J. 66 (1992), no. 2, 169–206.

Let $B_n=\sup_\Sigma |b_n|$. The question is about the order of decrease of $B_n$.
An easy estimate is $B_n\leq n^{-1/2}$, and the old result of Clunie and Pommerenke
says $B_n\leq Cn^{-0.503}$. I am not sure whether this has been ever improved, if it was, then by very little. Carleson and Jones proved that $\gamma:=-\lim \log B_n/\log n$ exists. (So by Clunie-Pommerenke, $\gamma>0.5$). The extremal functions are apparently conformal maps onto complements of some Julia sets, or other self-similar fractals. Using  Julia sets, Carleson and Jones were able to show
that $\gamma\leq 0.79$,( computer-assisted), and conjectured that in fact $\gamma=3/4$. Probably this conjecture is unpublished but it has been discussed among the specialists in the 1990s. I don't know of any substantial progress since then.