Asymptotic nonintegrality conjecture for the affine linear group intersection density

Let $q$ be a prime power, and let $\operatorname{AGL}(2,q)$ act on the lines of the affine plane $\operatorname{AG}(2,q)$. Write $\rho(\operatorname{AGL}(2,q))$ for the intersection density of this action.

Affine intersection-density conjecture. For every $\varepsilon>0$, there exists a prime power $q_0$ such that for every prime power $q\geq q_0$,

$$0\leq \rho(\operatorname{AGL}(2,q))-1\leq\varepsilon.$$

In particular,

$$\rho(\operatorname{AGL}(2,q))\in\mathbb{Q}\setminus\mathbb{N}$$

for every prime power $q$.

The conjecture is based on observed intersection densities for $q\in\{3,4,5,7,8\}$. It predicts convergence to $1$ from above together with nonintegrality, but the supplied text gives no resolution.