The conjecture on energy boundedness and decay for Maxwell fields near-extremal Kerr

Fix $M>0$. Let $\eta\in(0,1)$ and $m\in\mathbb{Z}$. Let $J_2,J_3\geq J_1\geq J_{\mathrm{min}}\geq 1$, and let $C=C(m,M,J_1,J_2,J_3,\eta)>0$. For any $|a|\leq M$ and any solution {\boldsymbol}{\mathfrak S}=({\boldsymbol}{\alpha}[{\boldsymbol}{\mathfrak S}],{\boldsymbol}{\underline{\alpha}}[{\boldsymbol}{\mathfrak S}],\widehat{{\boldsymbol}{\rho}}[{\boldsymbol}{\mathfrak S}],\widehat{{\boldsymbol}{\sigma}}[{\boldsymbol}{\mathfrak S}]) to the modified Maxwell equations arising from seed initial data supported on a fixed azimuthal mode $m$ on $\Sigma_0$, let $\mathbb{D}_{\mathrm{bdd}}^J$ and $\mathbb{D}_{\mathrm{dec}}^J$ denote the corresponding $J$-th order energy quantities depending only on the seed initial data. Sub-extremal and extremal cases. The stated degenerate energy boundedness, degenerate energy decay, and event-horizon decay estimates hold: for all $\tau\geq0$, the two degenerate energies are bounded by $C$ times their initial $\mathbb{D}_{\mathrm{bdd}}^{J_2}$ quantities; the two degenerate energies decay at rate $(1+\tau)^{-(1-\eta)}$ with initial $\mathbb{D}_{\mathrm{dec}}^{J_3}$ quantities; and the event-horizon energy and the two displayed angular-derivative quantities decay respectively at rates $(1+\tau)^{-(2-\eta)}$ and $(1+\tau)^{-(1-\eta)}$, with the corresponding initial $\mathbb{D}_{\mathrm{dec}}^{J_3}$ quantities. This is the expected uniform extension of energy boundedness and weaker decay to the nearly extremal regime, including the extremal endpoint; the preceding discussion explains that the established sub-extremal estimates degenerate as $|a|\to M$, while the conjectured estimates are motivated by expected control of the extremal Maxwell components under additional assumptions on the initial data.