The bilinear sparse square-function domination conjecture

Let $T$ be a Calderón–Zygmund operator on $\mathbb{R}^d$, and let $f,g$ be measurable bounded compactly supported functions on $\mathbb{R}^d$. For a sparse family $\mathcal{S}$, let $\mathcal{A}_{\mathcal{S},2}$ denote the sparse square function

$$\mathcal{A}_{\mathcal{S},2}f=\left(\sum_{Q\in\mathcal{S}}\langle|f|\rangle_Q^2 1_Q\right)^{1/2}.$$

Bilinear sparse square-function domination conjecture. There exist $\eta=\eta(d,T)\in(0,1)$ and an $\eta$-sparse family $\mathcal{S}$ of cubes in $\mathbb{R}^d$, depending on $d,T,f,g$, such that

$$|\langle T(f),g\rangle|\le C\left(\langle\mathcal{A}_{\mathcal{S},2}f,|g|\rangle+\langle|f|,\mathcal{A}_{\mathcal{S},2}g\rangle\right),$$

where $C=C(d,T)$ depends only on $d,T$. This is presented as a stronger form of sparse domination for Calderón–Zygmund operators; the supplied text gives no resolution.