Hytönen–Lacey mixed $A_p$–$A_\infty$ estimate with one supremum

Let $1<p<finfty$, let $T$ be a Calderon-Zygmund operator, and let $w otin A_p$. Write $p'=p/(p-1)$ and let $\sigma=w^{-1/(p-1)}$ be the dual weight. For mixed constants, $[w]_{(A_p)^a(A_\infty)^b}$ denotes the corresponding one-supremum mixed $A_p$-$A_\infty$ characteristic. Hytoten-Lacey conjecture. One should have

$$\|T\|_{L^p(w)}\lesssim [w]_{(A_p)^{\frac{1}{p}}(A_\infty)^{\frac{1}{p'}}}+[\sigma]_{(A_{p'})^{\frac{1}{p'}}(A_\infty)^{\frac{1}{p}}}.$$

The conjecture seeks a one-supremum mixed estimate improving the earlier mixed bounds. The paper gives only partial results, with additional logarithmic factors, and does not settle whether those factors are necessary.