Pseudoreflection-group classification conjecture

Let $(G,V)$ be a faithful, irreducible module of finite Morley rank, with $G$ connected. A pseudoreflection subgroup is an abelian subgroup $R$ such that

$$V=[V,R]\oplus C_V(R),$$

and $R$ acts transitively on the nontrivial elements of $[V,R]$. Pseudoreflection-group conjecture. There is a definable field $\mathbb{K}$ such that $(R,[V,R])$ is definably equivalent to $(\mathbb{K}^{\times},\mathbb{K}_+)$, and $(G,V)$ is definably equivalent to $(\operatorname{GL}(V),\mathbb{K}^n)$. The case $\operatorname{rk}[V,R]=1$ is especially important and is the only case needed for the proposed deduction of the highly generically transitive module conjecture; significant progress is known, but the full statement remains open.