The linear weighted bound for rough homogeneous singular integrals

Let $T_{\Omega}$ be the rough homogeneous singular integral associated with an angular function $\Omega$, let $w\in A_2$ be a Muckenhoupt weight, and let $[w]_{A_2}$ denote its classical $A_2$ characteristic. Linear $A_2$ conjecture. For every $w\in A_2$, one has

$$\|T_{\Omega}f\|_{L^2(w)}\le c_d\|\Omega\|_{L^\infty}[w]_{A_2}\|f\|_{L^2(w)}.$$

The conjecture predicts that the extra weight factor in the available quantitative estimate is unnecessary. It is stated for the $p=2$ case of rough homogeneous singular integrals and would give linear dependence on the $A_2$ characteristic.