Conjecture on sublinear transition-zone width in all dimensions

Let $u_\omega$ be the solution from Definition, under the hypotheses of Theorem (i) but without its limitation on the dimension $d$, and let $L_{u_\omega,\eta}(t)$ denote the transition-zone width for $\eta>0$. Sublinear transition-zone conjecture. For each $\eta>0$ and almost all $\omega\in\Omega$,

$$\lim_{t\to\infty}\frac{1}{t}L_{u_\omega,\eta}(t)=0.$$

If true, this would extend the dimension-dependent results to all dimensions: together with Theorem 1.8(i), it would imply the existence of a deterministic Wulff shape without the restriction on $d$. The source explains that known counterexamples in dimensions $d\geq 4$ rely on reaction properties occurring with probability zero in the stationary ergodic setting, so the assertion remains plausible but open.