Stationary fixed-point conjecture for the Rath–Toth forest fire equations

Let $v_k^n(t)$ be the random proportion of vertices at time $t$ that belong to clusters of size $k$ under the stationary law $\mathbb{P}_{\mathrm{stat}}^n$, and let $(w_k)_{k=1}^{\infty}$ be the fixed-point solution of the critical forest fire equations. Stationary fixed-point conjecture. For every $\epsilon>0$ and $T>0$,

$$\mathbb{P}_{\mathrm{stat}}^n\left(\sup_{t\in[0,T]}\sup_{k\geq 1}\left|v_k^n(t)-w_k\right|>\epsilon\right)\to 0\qquad\text{as }n\to\infty.$$

This asserts uniform-in-time and uniform-in-cluster-size convergence in probability of the stationary states to the critical fixed point. The source states that this convergence is not known and that existing results do not apply because the fixed point fails the required third-moment condition.