Conjecture on the limiting density of the sets $V_n$

Let $V_n$ be the related set of maps introduced in the paper, with asymptotic density $D(V_n)$, and define

$$L=\lim_{n\to\infty}D(V_n).$$

The paper establishes

$$0\leq L\leq e^{-1}.$$

Limiting $V_n$-density conjecture. The limiting density is

$$\lim_{n\to\infty}D(V_n)=\frac{1}{e}.$$

This conjecture asserts equality in the paper's asymptotic upper bound and is motivated by experimental data. No resolution is given in the supplied text.