Strong-solution convergence conjecture for one-dimensional coarsening

Let $(\Omega,\mathbb P)$ be the probability space of initial configurations, and let $\Psi_t^{(L)}(N)$ denote the evolution at time $t$ in the finite-volume system of size $L$. Strong-solution conjecture. For $\mathbb P$-almost every $N\in\Omega$, the limit

$$\Psi_t(N):=\lim_{L\to\infty}\Psi_t^{(L)}(N)$$

exists and defines a trajectory $t\mapsto\Psi_t(N)$. Together with the stated limiting lemma, this would yield a strong solution to the evolution rule; its proof is suggested as a plausible consequence of flux estimates.