The higher-dimensional energy condition conjecture for strongly gradient elliptic singular integrals

Let $T$ be a vector of standard singular integrals in higher dimensions, and suppose that $T$ is both strongly elliptic and strongly gradient elliptic. The energy conditions are the corresponding two-weight energy inequalities for $T$ and its dual. Energy condition conjecture. The energy conditions are necessary for the boundedness of $T$ on the relevant two-weight $L^{2}$ spaces. If true, the $T1$ theorem would follow for such operators from the cited main theorem. This conjecture is presented in a paper whose main counterexample disproves the analogous necessity statement for a small elliptic perturbation of the Hilbert transform; the higher-dimensional strongly gradient elliptic case is not resolved here.