Almost-sure scaling conjecture for Betti numbers in the Eden model

Let $\beta_i(t)$ be the $i$th Betti number of the $d$-dimensional Eden growth model at time $t$, with $1\leq i\leq d-1$. Almost-sure scaling conjecture. There exists a constant $C_{i,d}>0$ such that

$$\frac{\beta_i(t)}{t^{\frac{d-1}{d}}}\longrightarrow C_{i,d}$$

almost surely as $t\to\infty$. This strengthens the known high-probability growth estimates and is suggested by computational experiments; the asserted almost-sure convergence is open.