The asymptotic Poncelet n-gon proportion conjecture for conic pairs

Let $q$ be a prime power, and let $\tau_n$ denote a constant associated with the Poncelet $n$-gon condition. Consider pairs of conics in $\mathbf P^2(\mathbb F_q)$ satisfying that condition.

Poncelet n-gon proportion conjecture. The proportion of conic pairs in $\mathbf P^2(\mathbb F_q)$ satisfying the Poncelet $n$-gon condition is asymptotically equal to

$$\frac{\tau_n}{q},$$

for some integer value $\tau_n$.

The paper notes that $\tau_3=1$ follows from its main theorem. The conjecture is motivated by computational experiments, but the general asymptotic statement and the values of $\tau_n$ remain open.