Existence conjecture for the frog-model phase transition on transitive graphs

Let $G=(V,E)$ be a transitive graph with superlinear growth. For $x uin V$, let $B_x(n)$ denote the ball of radius $n$ centered at $x$, and write $g_x(n)=|B_x(n)|$. The graph has superlinear growth if $\limsup_{n\to\infty}g(n)/n=+\infty$, where $g(n)=g_{\mathbf{0}}(n)$. Let $\lambda_{\mathrm{c}}(t)$ and $t_{\mathrm{c}}(\lambda)$ be the critical particle-density and lifespan parameters of the frog model. Existence conjecture. For every $t>0$, $0<\lambda_{\mathrm{c}}(t)<\infty$, and for every $\lambda>0$, $0<t_{\mathrm{c}}(\lambda)<\infty$. This conjecture was resolved through the joint efforts of several works, especially the work cited in the source as PercolationPhaseTran.