Asymptotic average genus conjecture for rational knots and links

Let $K_n$ and $L_n$ denote, respectively, the sets of oriented rational knots and oriented rational links with crossing number $n$. For an object in either set, let $g$ be its genus, and let $\langle g\rangle_{K_n}$ and $\langle g\rangle_{L_n}$ denote the corresponding average genera.

Asymptotic average genus conjecture.

$$\lim_{n\to \infty}\frac{\langle g\rangle_{K_n}}{n}=\lim_{n\to \infty}\frac{\langle g\rangle_{L_n}}{n}=\frac{1}{4}.$$

The conjecture is proposed on the basis of strong numerical evidence, with the plotted averages appearing nearly linear in the crossing number. The preceding discussion establishes only the upper bounds $\langle g\rangle_{K_n}<n/2$ and $\langle g\rangle_{L_n}<n/2$; the asserted limiting value remains unresolved here.