Nam–Nguyen–Sly critical-value conjecture for configuration graphs

Let $\mu$ be a probability distribution on the non-negative integers satisfying

$$\sum_{k\geq 0}k(k-2)\mu(k)>0.$$

Let $(\mathscr{G}_n)_{n\in\mathbb{N}}$ be configuration graphs on $n$ vertices derived from $\mu$, and let $\mathscr{C}_n$ be the maximal component of $\mathscr{G}_n$. Write $\mathscr{T}$ for the local limit of $(\mathscr{C}_n)_{n\in\mathbb{N}}$, namely the unimodular Bienaymé–Galton–Watson tree associated with $\mu$ conditioned on non-extinction. Nam–Nguyen–Sly's critical-value conjecture. The three critical values coincide:

$$\lambda_-\big((\mathscr{C}_n)_{n\in\mathbb{N}}\big)=\lambda_+\big((\mathscr{C}_n)_{n\in\mathbb{N}}\big)=\lambda_{\mathsf 1}(\mathscr{T}).$$

This conjecture concerns the phase transition and extinction-time behavior of the contact process on the giant component of configuration graphs with a supercritical degree distribution. The cited work of Nam, Nguyen and Sly motivates the conjecture through polynomial extinction times in the subcritical phase; its resolution is not established in the supplied context.