The higher-dimensional fractional $T1$ conjecture

Let $\mathbf{T}^{\alpha,n}$ denote an elliptic vector of standard $\alpha$-fractional singular integrals in $\mathbb{R}^{n}$, and let $\sigma$ and $\omega$ be measures on $\mathbb{R}^{n}$. The fractional $T1$ conjecture. The operator $\mathbf{T}^{\alpha,n}$ is bounded from $L^{2}(\mathbb{R}^{n};\sigma)$ to $L^{2}(\mathbb{R}^{n};\omega)$, namely

$$\left\|\mathbf{T}^{\alpha,n}(f\sigma)\right\|_{L^{2}(\mathbb{R}^{n};\omega)}\lesssim \left\|f\right\|_{L^{2}(\mathbb{R}^{n};\sigma)},\qquad f\in L^{2}(\mathbb{R}^{n};\sigma),$$

if and only if the two one-tailed $\mathcal{A}_{2}^{\alpha}$ conditions with holes, the punctured $A_{2}^{\alpha}$ conditions, and the two testing conditions hold:

$$\left\|\mathbf{T}^{\alpha,n}\mathbf{1}_{Q}\sigma\right\|_{L^{2}(\mathbb{R}^{n};\omega)}\lesssim \sqrt{|Q|_{\sigma}},$$ $$\left\|\mathbf{T}^{\alpha,n,\operatorname{dual}}\mathbf{1}_{Q}\omega\right\|_{L^{2}(\mathbb{R}^{n};\sigma)}\lesssim \sqrt{|Q|_{\omega}},$$

for all cubes $Q$ in $\mathbb{R}^{n}$, whose sides need not be parallel to the coordinate axes. The conjecture seeks a complete two-weight characterization for higher-dimensional fractional singular integrals; the supplied text says that earlier results required energy conditions, but gives no resolution status for this full statement.