Fast-update limit conjecture for the critical infection rate

Let $q$ be fixed and assume that the existence assumption holds. Let $\tau_c(\text{CPDLP})$ denote the critical infection rate of the contact process on dynamical long-range percolation, and let $\tau_c^{\infty}(q)$ denote the critical infection rate of the corresponding contact process with infection kernel $(\lambda q p_e)_{e\in\mathcal{E}}$.

Fast-update limit conjecture.

$$\lim_{\gamma\to\infty}\lambda_c(\gamma,q)=\lambda^{\infty}_{c}(q).$$

The conjecture asserts that, when the edge-update speed tends to infinity, the critical infection rate converges to that of the limiting long-range contact process. The surrounding discussion establishes an upper bound on the limiting critical rate and notes that the comparison process converges to the limiting process from below, but does not resolve equality.