Continuity conjecture for the frog-model phase transition

Let $G$ be a transitive graph, and let the frog model on $G$ have critical curves $\lambda_{\mathrm{c}}(t)$ and $t_{\mathrm{c}}(\lambda)$. Continuity conjecture. The phase transition is continuous for all transitive graphs. The conjecture is motivated by the analogy with nearest-neighbor and finite-range Bernoulli percolation, but the source does not state a resolution.