Conjecture on asymptotic stabilization of the two-species regions

Let $R(t)$ and $B(t)$ be the red and blue regions of the two-species competition model, and let $\mathcal{S}$ be the Richardson-model limit shape. For a subset $Z\subset\mathbb R^d$, define

$$\hat{Z}=\{x\in\mathbb R^d:\operatorname{dist}(x,Z)\leq 1/2\},$$

where distance is measured in the $L^\infty$ norm, and for $s>0$ write $Z/s=\{y/s:y\in Z\}$. Asymptotic stabilization conjecture. There exist random sets $\tilde{R}$ and $\tilde{B}$ such that, with probability one,

$$\lim_{t\to\infty}\hat{R}(t)/t=\tilde{R},$$ $$\lim_{t\to\infty}\hat{B}(t)/t=\tilde{B},$$

and

$$\tilde{R}\cup\tilde{B}=\mathcal{S}.$$

The conjecture asserts that the red and blue regions have deterministic-scale limiting shapes whose union is the Richardson limit shape; the source gives simulations as evidence, but does not establish these limits.