Conjecture on forbidden intervals of irredundant-base cardinalities for primitive groups

Let $\mathcal{I}(G,\Omega)$ denote the set of cardinalities of irredundant bases of a permutation group $G$ on $\Omega$. Forbidden-interval conjecture. There exists an interval $X$ of positive integers, with $1\notin X$, such that no primitive permutation group $G$ on $\Omega$ satisfies

$$X=\mathcal{I}(G,\Omega).$$

This conjecture asserts that at least one interval excluding $1$ cannot arise as the set of irredundant-base cardinalities of a primitive permutation group. The source presents it as an open question motivated by the lack of a stronger realization conjecture for all such intervals.