Non-realizability conjecture for the finite-field arrangement with $2q$ hyperplanes

Let $q$ be a prime power with $q\notin\mathbb{P}$, and let $\mathcal{A}$ be the arrangement with $2q$ hyperplanes defined in the source's Definition n0series. Two arrangements have the same incidence when their incidence structures of hyperplanes and intersections are isomorphic.

Non-realizability conjecture. There is no arrangement in $\mathbb{C}^3$ with the same incidence as $\mathcal{A}$.

The arrangement is realizable over the relevant finite field when $q$ is an odd prime, as established immediately before this conjecture. The proposed statement concerns prime powers that are not prime and asserts that their incidence cannot be realized over $\mathbb{C}$; the source supplies no resolution.