Trivial-critical-parameter conjecture for graphs with $p_{\mathrm{c}}(G)=1$

Let $G$ be a fixed graph, and let $p_{\mathrm{c}}(G)$ denote its Bernoulli percolation critical parameter. Assume $p_{\mathrm{c}}(G)=1$, and let $\lambda_{\mathrm{c}}(t)$ and $t_{\mathrm{c}}(\lambda)$ be the frog-model critical parameters. Trivial-critical-parameter conjecture. For every $\lambda,t>0$, one has $t_{\mathrm{c}}(\lambda)=\infty$ and $\lambda_{\mathrm{c}}(t)=\infty$. The source gives no resolution.