Positive critical infection-rate conjecture for dynamical long-range percolation

Let $\mathcal{E}$ be the edge set, and let $(v_e)_{e\in\mathcal{E}}$ and $(p_e)_{e\in\mathcal{E}}$ be the edge-update speeds and edge probabilities. Assume that they satisfy the existence assumption. Let $\lambda_c(\gamma,q)$ be the critical infection rate, with update parameter $\gamma>0$ and percolation parameter $q\in[0,1]$.

Positive critical-rate conjecture. For every $\gamma>0$ and $q\in[0,1]$,

$$\lambda_c(\gamma,q)>0.$$

Thus the model should have a subcritical phase for every choice of the update speed and percolation parameter. The discussion explains that standard branching-process comparisons may fail on locally finite random graphs with unbounded degree, so positivity of the critical rate is not automatic; the conjecture remains open in the stated generality.