The separated bumps conjecture for Calderón-Zygmund operators

Let $\sigma$ and $w$ be two weights with densities, and let $1<p<\infty$. Let $A\in B_p$ and $B\in B_{p'}$. For a Calderón-Zygmund operator $T$, write $T_\sigma f=T(\sigma f)$, and let $[\sigma,w]_{\overline A,p'}$ and $[w,\sigma]_{\overline B,p}$ denote the corresponding two-weight bump characteristics. The constant $C_T$ is the constant defined in the source.

Separated bumps conjecture. For all Calderón-Zygmund operators $T$, there holds

$$\lVert T_\sigma:L^p(\sigma)\to L^p(w)\rVert\lesssim C_T\bigl\{[\sigma,w]_{\overline A,p'}+[w,\sigma]_{\overline B,p}\bigr\}.$$

The implied constant depends upon the choice of $A$ and $B$.

This conjecture asks whether the two bump conditions can occur separately, rather than in the same product as in the preceding sufficient condition. The supplied text attributes it to Cruz-Uribe, Reznikov, and Volberg; its resolution is not specified in the source.