The survival probability of a heavy-tailed critical Derrida-Retaux system

For the Derrida–Retaux recursion $X_{n+1}=(X_n^{(1)}+\dots+X_n^{(m)}-1)^+$ in the critical regime $(m-1)\mathbb{E}(X_0m^{X_0})=\mathbb{E}(m^{X_0})<\infty$, Chen, Derrida, Hu, Lifshits and Shi conjectured that $\mathbb{P}(X_n\ne0)\sim 4(m-1)^{-2}n^{-2}$. We give a nonconstant critical law for $m=2$, with $\mathbb{P}(X_0=k)=c\,2^{-k}k^{-3}$ for $k\ge2$, for which $\liminf_n n^2\,\mathbb{P}(X_n\ne0)\le 3/2$. So the conjecture does not hold under the critical moment condition alone. The bound $3/2$ is the constant that Chen, Hu and Shi predict for this tail. The proof combines an exact recursion for $\mathbb{E}(2^{X_n})$ with the upper bound of Chen and Shi on the product $\prod_{i<n}\mathbb{E}(2^{X_i})$.