Upper decay-bound conjecture for NIMFA SIS prevalence below threshold

Let $G$ be a graph, let $K_N$ be the complete graph on $N$ vertices, let $u$ denote the all-infected initial infection-probability vector, and let $V(0)$ be an initial infection-probability vector. Write $y(t;G,\tau,V(0))$ for the prevalence of the epidemic process on $G$ with effective infection rate $\tau$, and let $\tau_c^{(1)}(G)$ denote the epidemic threshold. Upper decay-bound conjecture. If $\tau\leq\tau_c^{(1)}(G)$, then

$$y(t;G,\tau,V(0))\leq y(t;G,\tau_c^{(1)}(G),u)\leq y(t;K_N,\tau_c^{(1)}(K_N),u)=\frac{1}{1+t}.$$

The formula for the complete graph is proved in the cited lemma, while the displayed upper bound is supported analytically in the appendix; the parser marks this conjecture as resolved.