Poncelet polygon density conjecture for pairs of conics

Let $FbbF_q$ be a finite field of characteristic greater than $3$, and let $n$ be an integer coprime to $q$. Let $\Gamma_n$ denote the set of ordered pairs of smooth conics satisfying the Poncelet $n$-gon condition, and let $\Psi$ denote the space of ordered pairs of smooth conics. For an integer $n$, write $d'(n)$ for the number of divisors of $n$ other than $1$ and $2$.

Poncelet polygon density conjecture. The density satisfies

$$\frac{|\Gamma_n|}{|\Psi|}=\frac{d'(n)}{q}+O(q^{-3/2}).$$

This conjecture extends the proved odd-$n$ asymptotic to even integers, including a corrected version of Chipalkatti's tetragon conjecture. The preceding results establish the corresponding asymptotic for odd $n$ coprime to $q$, while the general case remains open.