Cameron's Greedy Conjecture for primitive permutation groups

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite permutation group. A base for $G$ is a sequence of points of $\Omega$ with trivial pointwise stabiliser, and $b(G)$ denotes the minimum size of a base. A greedy base is constructed by choosing each successive point in a longest orbit of the pointwise stabiliser of the previously chosen points; let $\mathcal{G}(G)$ be the largest size of a greedy base for $G$. Cameron's Greedy Conjecture. There is an absolute constant $c$ such that

$$\mathcal{G}(G)\leqslant cb(G)$$

for every finite primitive group $G$. The conjecture would improve the general bound $\mathcal{G}(G)\leqslant db(G)\log\log|\Omega|$ to a constant-factor bound for primitive groups; it was proposed by Cameron in 1999 and its status is not resolved by the supplied source.