Boundary convergence conjecture for type II parabolic self-maps

Let $\phi$ be an analytic self-map of the unit disk $\mathbb{D}$, and write $\phi_n=\phi^{\circ n}$ for its $n$-th iterate. Suppose that $\phi$ is type II parabolic, meaning that its hyperbolic steps tend to zero, and let $p$ be its Denjoy–Wolff point. Boundary convergence conjecture. The sequence $\phi_n^{\star}(\zeta)$ converges to $p$ for almost every $\zeta\in\partial\mathbb{D}$ if and only if $\phi$ is not an inner function. The preceding theorem establishes the analogous assertion for elliptic, hyperbolic, and type I parabolic self-maps; the type II parabolic case is left open because the authors' methods do not extend to it and it is described as more similar to the elliptic case.