Conjecture on realizing minimal-base cardinalities for transitive permutation groups

Let $X\subseteq\mathbb{N}$ with $1\notin X$. For a permutation group $G$ acting transitively on a domain $\Omega$, let $\mathcal{M}(G,\Omega)$ denote the set of cardinalities of minimal bases of $G$ on $\Omega$. Realization conjecture. There exists a transitive permutation group $G$ on $\Omega$ such that

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

The conjecture proposes that every subset of the natural numbers not containing $1$ can occur as the set of minimal-base cardinalities for a transitive permutation group. The source gives examples for certain non-interval subsets but notes that, for example, $\{4,6\}$ is not known to be realizable.