Infinitely many connected components in a Master Teapot slice

Let $\Xi_\lambda$ denote the $\lambda$-slice of Thurston's Master Teapot, and let $\overline{\mathbb{D}}$ be the closed unit disk. Connected-component conjecture. There exists $\lambda\in(1,2)$ such that

$$\Xi_\lambda\cap\overline{\mathbb{D}}$$

has infinitely many connected components. This is presented as a consequence that would follow if the preceding asymptotic Julia–Mandelbrot correspondence were true, or at least true for sufficiently many points $z$; its resolution is not supplied in the source.