The NTV conjecture for the Hilbert transform

Let $b$ and $b$ be weights on $b^n$, let $H$ be the Hilbert transform, and let $b_Q$ denote the two-tailed auxiliary function associated with a cube $Q$. Write $b_Q$ and $b_Q$ for the $b$- and $b$-measures of $Q$, respectively. NTV conjecture. The Hilbert transform is bounded from $L^2(\mathbb{R}^n;\sigma)$ to $L^2(\mathbb{R}^n;\omega)$, namely

$$\left\Vert H(f\sigma)\right\Vert_{L^2(\mathbb{R}^n;\omega)}\lesssim \left\Vert f\right\Vert_{L^2(\mathbb{R}^n;\sigma)},\qquad f\in L^2(\mathbb{R}^n;\sigma),$$

if and only if the two-weight $A_2$ condition with two tails holds,

$$\left(\frac{1}{|Q|}\int_Q \mathbf{s}_Q^2\,d\omega(x)\right)\left(\frac{1}{|Q|}\int_Q \mathbf{s}_Q^2\,d\sigma(x)\right)\lesssim 1,$$

uniformly over all cubes $Q$, and the testing conditions

$$\left\Vert H\mathbf{1}_Q\sigma\right\Vert_{L^2(\mathbb{R}^n;\omega)}\lesssim \sqrt{|Q|_\sigma},\qquad \left\Vert H^*\mathbf{1}_Q\omega\right\Vert_{L^2(\mathbb{R}^n;\sigma)}\lesssim \sqrt{|Q|_\omega}$$

hold uniformly over all cubes $Q$. This is the two-weight characterization sought for the Hilbert transform, extending the positive fractional-integral theory to a nonpositive singular kernel; the source does not state a resolution.