The general $T1$ conjecture for elliptic fractional singular integrals

Let $b^{\alpha,n}$ denote an elliptic vector of standard $b$-fractional singular integrals in $b^n$, and let $b$ and $b$ be weights. The notation $\mathcal{A}_2^\alpha$ denotes the one-tailed Muckenhoupt condition with holes, $A_2^{\alpha,\mathrm{punct}}$ the punctured condition, and $b^{\alpha,n,\mathrm{dual}}$ the dual vector. General $T1$ conjecture. The operator $b^{\alpha,n}$ is bounded from $L^2(\mathbb{R}^n;\sigma)$ to $L^2(\mathbb{R}^n;\omega)$, namely

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

if and only if the two one-tailed $\mathcal{A}_2^\alpha$ conditions with holes, the punctured $A_2^{\alpha,\mathrm{punct}}$ conditions, and the testing conditions

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

hold 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 $T1$ theorem for higher-dimensional and fractional singular integrals. The cited prior result established such a theorem under energy side conditions, so the asserted equivalence remains open in the source.