Non-vanishing charge conjecture for generic black holes

Consider smooth developments of the Einstein–Maxwell–Klein–Gordon system as in Theorem 1.1, with coupling parameter $\mathfrak{e}\neq 0$. Let $\mathcal{Q}^+$ be the quotient spacetime, $J^-(\mathcal{I}^+)$ the causal past of future null infinity, and $Q_+$ the asymptotic charge of the black hole. Non-vanishing charge conjecture. Among all the data admissible from Theorem 1.1 with $\mathfrak{e}\neq 0$, there exists a generic sub-class for which, if the maximal future development has

$$\mathcal{Q}^+\backslash J^-(\mathcal{I}^+)\neq\emptyset,$$

then the asymptotic charge $Q_+$ of the black hole is non-zero.

The conjecture asserts that charged black holes are generically not completely neutralized by radiation to infinity. The supplied text gives no resolution.