Above-threshold prevalence-distance decay conjecture for NIMFA SIS epidemics

Let $G$ be a graph, let $K_N$ be the complete graph on $N$ vertices, let $u$ be the all-infected initial infection-probability vector, and let $V_\infty$ be the endemic steady-state infection-probability vector. Write $y(t;G,\tau,V(0))$ for the prevalence with effective infection rate $\tau$, and let $y_\infty(G;\tau)$ denote its steady-state prevalence. Above-threshold decay conjecture. For any $\tau>\tau_c$, if $V(0)\geq V_\infty$, then

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

Thus convergence to the endemic steady state is conjectured to be no slower than convergence of the critical complete-graph process to $y_\infty=0$. The supplied text gives no resolution evidence.