Conjecture on the distributional convergence of rescaled UIPT ball volumes

Let $B_r$ be the ball of radius $r$ around the root in the type II uniform infinite planar triangulation, and let $\overline{B}_r$ be its hull, obtained by adjoining to $B_r$ all finite components of its complement. The notation $|T|$ denotes the number of vertices in a triangulation $T$. Distributional convergence conjecture. The random variables

$$r^{-4}|B_r| \quad\text{and}\quad r^{-4}|\overline{B}_r|$$

converge in distribution as $r\to\infty$. The preceding growth estimates and the methods used to prove them suggest that these rescaled volumes have limiting distributions, but the statement does not specify those distributions or establish convergence.