Asymptotic density conjecture for odd toroidal covering parameters

Let $\xi(n)$ denote the toroidal covering parameter, and let $n$ range over positive integers. Asymptotic density conjecture.

$$\lim_{n\rightarrow\infty}\frac{\xi(2n+1)}{2n+1}=\frac{2}{3}.$$

The conjecture predicts the limiting density of the parameter along odd arguments. It is presented without a proof or resolution in the paper.