Base-size conjecture for definably primitive groups of finite Morley rank

Let $(G,X)$ be a faithful definable permutation group of finite Morley rank, let $G$ be connected and definably primitive on $X$, and let $b(G)$ denote the minimal cardinality of a base. Write $\operatorname{rk}(X)$ for the Morley rank of $X$. Base-size conjecture. There is a constant $c$, independent of $G$ and $X$, such that

$$b(G)<c\cdot\operatorname{rk}(X),$$

and the set of minimal bases is generic in $X^{b(G)}$, of rank $b(G)\cdot\operatorname{rk}(X)$. This would give uniform control of bases and hence of definable parametrisations of binding groups; the conjecture is presented as open.