Algebraicity characterization for rank-three Dowling geometries

Let $\Gamma$ be a nontrivial finite group, and let $p$ be a prime. Dowling-geometry algebraicity conjecture. The matroid $\operatorname{DG}(3,\Gamma)$ is algebraic over $\operatorname{GF}(p)$ if and only if there is some integer $j\ge 1$ such that $\Gamma$ is a subgroup of $\mathbb Z_{p^j-1}$. This is the analogous conjecture for Dowling geometries and would contribute to characterizing the fields over which these geometries are algebraic. The source does not give a resolution; the conjecture remains open.