Regular-cycle proportion conjecture for primitive permutation groups

Let $(G,\Omega)$ be a primitive permutation group not covered by the structural conjecture above. For $g\in G$, let $o_g$ denote the number of regular cycles of $g$, and let $c_g$ denote the total number of cycles of $g$, including cycles of length $1$.

Regular-cycle proportion conjecture. There exists an absolute positive constant $c$ such that, for every such $(G,\Omega)$ and every $g\in G$,

$$\frac{o_g}{c_g}>c.$$

This conjecture predicts a uniform positive proportion of regular cycles outside the exceptional structural family. It is presented in the paper as a further conjecture, and no resolution is supplied in the given text.