Scaling-limit conjecture for dynamical supercritical oriented percolation

Let $\Gamma_s^\epsilon$ be the set of right-most infinite open paths at dynamical time $s$. Let $S_{\alpha(p),\sigma(p),\epsilon}$ denote the corresponding space-time rescaling, and let $({\cal W}_s; s\geq 0)$ be the dynamical Brownian web. Scaling-limit conjecture. There exists a constant $c(p)$ such that

$$S_{\alpha(p),\sigma(p),\epsilon}(\Gamma_s^\epsilon; s\geq 0) \underset{\epsilon\to 0}{\Longrightarrow} ({{\cal W}_{c(p)s}}; s\geq 0).$$

This asserts that the dynamically evolving right-most infinite paths of supercritical oriented percolation converge, after the model's characteristic rescaling, to a time-rescaled dynamical Brownian web. The statement identifies the continuum dynamical object arising from perturbing the percolation environment, with the constant $c(p)$ accounting for the dynamical-time normalization.