Uniform iteration conjecture for essentially linear fractional self-maps of the disk

Let $\mathbb{D}$ be the open unit disk, and let $\varphi$ be an essentially linear fractional self-map of $\mathbb{D}$ with exactly one fixed point $w$ in $\overline{\mathbb{D}}$. Uniform iteration conjecture. Then $\varphi$ converges under iteration to $w$ uniformly on all of $\mathbb{D}$. This conjecture concerns the relationship between uniformly convergent iteration and essentially linear fractional self-maps. The paper gives examples showing that these two classes do not coincide, but notes that their intersection is non-trivial; the conjectured implication remains open in the supplied source.