DMS's conjecture on intervals of irredundant base cardinalities for primitive groups

Let $G$ be a finite permutation group acting on a domain $\Omega$. An irredundant base is an ordered sequence $(\omega_1,\ldots,\omega_\ell)$ whose associated stabilizer chain

$$G\ge G_{\omega_1}\ge G_{\omega_1,\omega_2}\ge \cdots \ge G_{\omega_1,\omega_2,\ldots,\omega_\ell}=1$$

has all inclusions strict, and define

$$\mathcal{I}(G,\Omega)=\{\ell\in\mathbb{N}\mid \text{there exists an irredundant base of length }\ell\}.$$

DMS's conjecture. There exists an interval $X$ of positive integers, not containing $1$, such that no primitive group $G$ acting on any domain $\Omega$ satisfies

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

Cameron proved that $\mathcal{I}(G,\Omega)$ is always an interval, while previous work established that every interval of natural numbers not containing $1$ can be realized by a finite transitive permutation group. The conjecture asserts that at least one such interval is impossible for primitive groups, distinguishing the primitive case from the transitive imprimitive constructions.