Asymptotic parameter limits in the Eden model for pentagons

Let $n$ be the number of pentagons in the model, and let $V(n)$, $E(n)$, and $H(n)$ denote the numbers of vertices, edges, and holes, respectively. The expectation is taken over the randomness of the Eden growth process. Asymptotic parameter conjecture. The expected values satisfy

$$\lim_{n\to\infty}\frac{\mathbb{E}[V(n)]}{n}\approx 2.68,\qquad \lim_{n\to\infty}\frac{\mathbb{E}[E(n)]}{n}\approx 4.02,\qquad \lim_{n\to\infty}\frac{\mathbb{E}[H(n)]}{n}\approx 0.34.$$

These conjectured limits are supported by numerical simulations. Euler's formula in this setting, $V-E+n+H=1$, relates the three quantities, but the source gives no proof of the stated asymptotic values.