Strict monotonicity conjecture for the frog-model critical curves

Let $G$ be a transitive graph of superlinear growth, and let $\lambda_{\mathrm{c}}(t)$ and $t_{\mathrm{c}}(\lambda)$ denote the critical curves of the frog model. Strict monotonicity conjecture. The maps $t\mapsto\lambda_{\mathrm{c}}(t)$ and $\lambda\mapsto t_{\mathrm{c}}(\lambda)$ are continuous and strictly decreasing. The source presents this as a natural question about the critical curves and gives no resolution.