The Genericity Conjecture for connected groups of finite Morley rank

Let $G$ be a connected group of finite Morley rank. A subset of $G$ is generic if it has full Morley rank in $G$.

Genericity Conjecture. The union of the connected definable nilpotent subgroups of $G$ contains a definable generic subset of $G$.

This is an outstanding open question concerning groups of finite Morley rank. In particular, it asks whether generic elements of a connected group are contained in connected definable nilpotent subgroups.