All-time proportionality conjecture for the wireless-network size estimator

Let $Z(t)$ be the number of nodes detected by the estimator at time $t$, let $\tilde{n}$ denote the BLUE of the network size, and let $N_{\max}$ be the total number of nodes. For coefficients $\alpha_k$, the estimator is conjectured to satisfy

All-time proportionality conjecture. The estimator is proportional to the BLUE and can be written as

$$\tilde{Z}(t)=N_{\max}\prod_{k=0}^{t}\alpha_k\,\tilde{n},$$

with

$$\prod_{k=0}^{\infty}\alpha_k=\frac{1}{N_{\max}} \qquad\text{and}\qquad \lim_{K\to N_{\max}}\prod_{k=0}^{t}\alpha_k=\frac{1}{N_{\max}}.$$

The claim extends the proportionality established in the preceding lemma for $t=0$, $t=1$, and $t=\infty$ to every time instant. Its validity beyond those cases is not established in the supplied text.