Outer-perimeter proportion conjecture for high-dimensional Eden models

For each dimension $d>1$, let $P_d(t)$ be the volume of the site perimeter of the $d$-dimensional Eden growth model, and let $\operatorname{OutP}_d(t)$ be the volume of its outer perimeter.

Outer-perimeter proportion conjecture. There exists a number $\operatorname{per}_d>0$ such that

$$\frac{\operatorname{OutP}_d(t)}{P_d(t)}\longrightarrow\operatorname{per}_d$$

almost surely as $t\rightarrow\infty$. Moreover,

$$\lim_{d\to\infty}\operatorname{per}_d=1.$$

Simulations suggest convergence to a dimension-dependent proportion, while available data are insufficient to estimate the limiting values reliably in dimensions three through five. The asserted convergence and high-dimensional limit remain unproved in the source.