Alternating-group rank conjecture for nonsolvable groups of finite Morley rank

Let $n$ be a positive integer, let $H$ be a nonsolvable connected group of finite Morley rank, and suppose that $\operatorname{Alt}(n)$ acts definably and faithfully on $H$ by automorphisms. Alternating-group rank conjecture. For all sufficiently large $n$,

$$n\leq \operatorname{rk} H.$$

The conjecture extends earlier work restricted to actions on groups of degenerate type. Its validity beyond that setting is posed as a key ingredient in the study of actions of alternating groups on nonsolvable groups of finite Morley rank.