Algebraicity characterization for rank-three lift geometries

Let $\Gamma$ be a nontrivial finite group, and let $p$ be a prime. Lift-geometry algebraicity conjecture. The matroid $\operatorname{LG}^+(3,\Gamma)$ is algebraic over $\operatorname{GF}(p)$ if and only if there is some integer $j\ge 1$ such that $\Gamma\cong \mathbb Z_p^j$. This would help characterize the fields over which lift geometries are algebraic. It is known that algebraicity over a field of characteristic $p$ forces every element of $\Gamma$ to have order a power of $p$, but whether $\Gamma$ must be Abelian is unclear.