The energy necessity conjecture for elliptic fractional singular integrals

Let $b^{\alpha,n}$ denote an elliptic vector of standard $b$-fractional singular integrals in $b^n$, acting between weighted spaces $L^2(\mathbb{R}^n;\sigma)$ and $L^2(\mathbb{R}^n;\omega)$. The energy conditions are the conditions defined later in the source. Energy necessity conjecture. If $b^{\alpha,n}$ is bounded from $L^2(\mathbb{R}^n;\sigma)$ to $L^2(\mathbb{R}^n;\omega)$, then the energy conditions hold. This conjecture identifies the energy conditions as necessary for two-weight boundedness of elliptic fractional singular integrals; the source presents it as stronger than the preceding general $T1$ conjecture, and gives no resolution.