The sparse square-function criterion for Calderón–Zygmund operators

Let $w,\sigma$ be weights on $\mathbb{R}^d$. For $1\le p<\infty$ and a sparse family $\mathcal{S}$, write

$$\mathcal{A}_{\mathcal{S},p}f=\left(\sum_{Q\in\mathcal{S}}\langle|f|\rangle_Q^p1_Q\right)^{1/p}.$$

Sparse square-function criterion. Suppose that for every $0<\eta<1$ and every $\eta$-sparse family $\mathcal{S}$ of cubes in $\mathbb{R}^d$,

$$\|\mathcal{A}_{\mathcal{S},2}(\mathord\cdot\,w)\|_{L^2(w)\to L^2(\sigma)},\quad \|\mathcal{A}_{\mathcal{S},2}(\mathord\cdot\,\sigma)\|_{L^2(\sigma)\to L^2(w)}\le C=C(\eta,d).$$

Then, for any Calderón–Zygmund operator $T$ on $\mathbb{R}^d$, the operator $T(\mathord\cdot\,w)$ is bounded from $L^2(w)$ into $L^2(\sigma)$. The conjecture proposes that uniform two-weight bounds for sparse square functions in both directions suffice for the corresponding singular-integral estimate; the supplied text does not state a resolution.