Asymptotic conjecture for smoothed connected essential-surface counts

Let $M$ be a manifold satisfying the hypotheses of Theorem~, and define the smoothed count

$$\abar_M(n)=\sum_{k\leq n}\atil_M(k),$$

where $\atil_M(n)=a_M(n+1)$. Asymptotic conjecture. Either $\atil_M(n)=0$ for all sufficiently large $n$, or there exists $s\in\mathbb N$ such that

$$\lim_{n\to\infty}\frac{\abar_M(n)}{n^s}$$

exists and is positive. This is motivated by computational plots and remains unproved in the supplied text.