Keith's dimension conjecture for the passant–internal incidence null space

Let $q$ be an odd prime power, let $\mathcal{O}$ be a conic in $\mathrm{PG}(2,q)$, and let $\mathbf{A}$ be the incidence matrix of passant lines and internal points with respect to $\mathcal{O}$. Let $\mathcal{L}$ be the column $\mathbb{F}_2$-null space of $\mathbf{A}$, and write $\dim_{\mathbb{F}_2}(\mathcal{L})$ for its dimension.

Keith's conjecture.

$$\dim_{\mathbb{F}_2}(\mathcal{L})=\frac{(q-1)^2}{4}.$$

The conjecture gives the dimension of the binary code arising from the passant-line versus internal-point incidence matrix. It is proved in the present paper, so the conjecture is solved.