Lacey's two-weight Hilbert transform conjecture with common point masses

Let

\sigma$ and $w$ be two Radon measures on

\mathbb{R}$. For each interval$I$, define the Poisson integral by

$$P(f\,\mathrm{d}\sigma,I):=\int_{\mathbb{R}}\frac{|I|f(x)\,\mathrm{d}\sigma(x)}{|I|^2+(x-c_I)^2}.$$

Let $b_I$ be the unique, if it exists, point such that

$$P(1_{\{b_I\}}\,\mathrm{d}\sigma,I)>\frac12P(\mathrm{d}\sigma,I),\qquad P(1_{\{b_I\}}\,\mathrm{d}w,I)>\frac12P(\mathrm{d}w,I).$$

Let $\widetilde\sigma_I:=\sigma-\sigma\{b_I\}\delta_{b_I}$ if $b_I$ exists, and $\widetilde\sigma_I:=\sigma$ otherwise; define $\widetilde w_I$ similarly. Lacey's conjecture. The Hilbert transform satisfies the norm inequality referenced in the source as

$$if and only if it satisfies the local testing conditions referenced as$$

and, for every interval $I$, the condition

$$P(\mathrm{d}\widetilde\sigma_I,I)P(\mathrm{d}w,I)+P(\mathrm{d}\widetilde w_I,I)P(\mathrm{d}\sigma,I)\leq ([\sigma,w]_{A_2}^{**})^2.$$

This conjecture proposes a complete characterization of the two-weight boundedness of the Hilbert transform, including the correction needed when the measures have common point masses; its status is not resolved by the supplied text.