The quadratic asymptotic conjecture for replication numbers of paths

Let $P_n$ be the path on $n$ vertices, and let $\rho_R(P_n)$ denote its replication number. The notation $O(n)$ means a quantity bounded in absolute value by a constant multiple of $n$ for all sufficiently large $n$. The quadratic asymptotic conjecture. For every $n\geq 1$,

$$\rho_R(P_n)=n^2/4+n^2/16+O(n).$$

The paper reports exact computations through $n=16$ and observes that the proven upper bound is close to these values; the displayed asymptotic assertion remains conjectural.