Characterization of bounded iterated commutators of the Hilbert transform

Let $b_1,b_2$ satisfy the same assumptions as in Theorems and. Let $H$ denote the Hilbert transform, let $A,B,C$ be Young functions, let $\bar A,\bar B,\bar C$ be their complementary Young functions, let $B_2$ be the relevant class of Young functions, and let $S_{A,B}(b_1,b_2)$ and $T_C(b_1,b_2)$ be the quantities defined in the paper. The boundedness characterization conjecture. The operator $[b_1,[b_2,H]]$ is bounded on $L^2(\mathbb{R})$ if and only if there exist Young functions $A,B,C$ such that $\bar A,\bar B,\bar C\in B_2$ and

$$S_{A,B}(b_1,b_2)+T_C(b_1,b_2)<\infty.$$

This conjecture seeks a necessary-and-sufficient condition for $L^2$ boundedness of the iterated commutator under the assumptions used for the paper's upper and lower bounds. The supplied text does not state whether the conjecture has been resolved.