Primitive homogeneous structures distinguishing-number conjecture

Let $\boldsymbol{S}$ be a countably infinite homogeneous structure, and let $D(\boldsymbol{S})$ denote the smallest size of a partition of its elements preserved only by the trivial automorphism of $\boldsymbol{S}$. The automorphism group of $\boldsymbol{S}$ is primitive when its action on the underlying set is primitive.

Primitive homogeneous structures conjecture. The distinguishing number of every primitive homogeneous countably infinite structure is two or infinite.

This conjecture extends the corresponding results for infinite homogeneous simple and directed graphs, where the distinguishing number is either two or infinite apart from structures with imprimitive automorphism groups. It is motivated by the finite theorem that sufficiently large finite primitive permutation groups have distinguishing number two, with known exceptions.