Boundedness conjecture for axisymmetric scalar waves in extremal Kerr interiors

Let $(\mathcal M,g)$ be extremal Kerr spacetime with mass $M$ and angular momentum per unit mass $a$, where $M=|a|\neq 0$. Let $\phi$ be an axisymmetric solution of

$$\Box_g\phi=0$$

arising from sufficiently regular Cauchy data on a hypersurface $\Sigma$, and let $V_-$ be regular at the Cauchy horizon $\mathcal C\mathcal H^+$. Extremal Kerr interior conjecture. Throughout the maximal domain of dependence $\mathcal D^+(\Sigma)$, one has

$$|\phi|\leq C,\qquad |\partial_{V_-}\phi|\leq C.$$

This is proposed by analogy with the extremal Reissner–Nordström conjecture and known horizon stability and instability results; the source explicitly does not venture a conjecture for non-axisymmetric solutions.