The one-sided Calderón–Zygmund weighted estimate conjecture

Let $T$ be a one-sided Calderón–Zygmund operator: an operator bounded on $L^2$ with a kernel $K(x,y)$ satisfying

$$\lvert K(x,y)\rvert \lesssim \lvert x-y\rvert^{-1},\qquad x\ne y,$$ $$\lvert \nabla^\alpha K(x,y)\rvert \lesssim \lvert x-y\rvert^{-1-\alpha},$$

for fixed $0<\alpha\leq 1$, and $K(x,y)=0$ when $x<y$. For $1<p<\infty$, let $p'$ be the conjugate exponent, let $w$ be a weight in $A_p^+$, and let $\sigma=w^{-1/(p-1)}$. The one-sided Calderón–Zygmund weighted estimate conjecture. Then

$$\lVert T\rVert_{L^p(w)\to L^p(w)}\lesssim [w]_{A_p^+}\max\left\{[\sigma]_{A_\infty^-}^{1/p},[w]_{A_\infty^+}^{1/p'}\right\}\lesssim [w]_{A_p}^{\max\{1,(p-1)^{-1}\}}.$$

This would be a one-sided analogue of weighted bounds for Calderón–Zygmund operators and of the main results cited in the source. The paper presents it as an overarching conjecture; its resolution is not specified in the supplied text.