Collinear-triple conjecture for permutation graphs

Let $\sigma:{\mathbb{F}_q}\rightarrow{\mathbb{F}_q}$ be a permutation, and let $\Gamma_\sigma=\{(x,\sigma(x)):x\in{\mathbb{F}_q}\}\subset{\mathbb{F}_q}^2$ be its graph. A collinear triple is a three-element subset of $\Gamma_\sigma$ whose points lie on one affine line. Collinear-triple conjecture. For any permutation $\sigma:{\mathbb{F}_q}\rightarrow{\mathbb{F}_q}$, the number of collinear triples in $\Gamma_\sigma$ is at least

$$\frac{q-1}{2}.$$

The source attributes this conjecture to an earlier work and notes that a lower bound of $(q-1)/4$ was already proved there; the stated $(q-1)/2$ bound is presented as the conjectural strengthening.