The finite Morley rank analogue of Glauberman's theorem for twisted subgroups

Let $G$ be a connected group of finite Morley rank, and let $L$ be a definable uniquely $2$-divisible twisted subgroup of Morley degree $1$ such that

$$G=\langle L\rangle.$$

Here, a twisted subgroup is a subset closed under inverses and under the operation $(x,y)\mapsto xyx$, and uniquely $2$-divisible means that every element has a unique square root in the relevant subgroup.

Finite Morley rank Glauberman conjecture. The group $G$ is solvable.

This is proposed as a finite Morley rank version of Glauberman's theorem, which proves solvability for groups generated by finite uniquely $2$-divisible twisted subgroups. The authors note that it is not known whether every uniquely $2$-divisible K-loop of finite Morley rank can be definably half-embedded into a group of finite Morley rank.