Theorem on the scaling limit for the pushed front

Let $\widehat{R}$ denote the front of the modified system in which, when multiple absorptions are possible, exactly one particle is absorbed, the front advances one step, and all excess particles are pushed one step to the right. Assume i.i.d. initial conditions and the critical parameter $\rho=1$ as in Theorem~. Scaling-limit conjecture. Theorem~ holds for $\widehat{R}$ in place of $R$. The modification avoids finite-time explosion even when $\rho>1$, and the conjecture proposes that the same critical scaling and limiting behavior established for $R$ also hold for the pushed front at $\rho=1$.