Briefly, my question is the following.
does every countable ω-categorical, ω-stable structure with disintegrated strongly minimal sets interpret in the countable pure set?
By countable pure set I mean a structure with countable universe and equality relation only.
This is a repetition of this question A totally categorical structure with trivial geometry which is not interpretable in the trivial structure. However, I do not understand why the answer provided there is marked correct. (I agree with Dima Sustretov in the comments that the structure does interpret in the pure set).
The background to my question is the following.
It is shown in the paper of Cherlin, Harrington and Lachlan that every ω-categorical, ω-stable structure is coordinatized by a collection of projective spaces, affine spaces and pure sets (which appear as strictly minimal sets in the expansion of the original structure by imaginaries). I'm interested in those structures in which only the pure sets appear. These were studied in the paper of Lachlan titled "Structures coordinatized by indiscernible sets". In this paper, it is shown that every such structure interpret in an arbitrary countable linear order, and also, that such structures correspond precisely to reducts of totally categorical structures with trivial geometry (of the strongly minimal sets).
It is easy to see that every structure which interprets in the pure set is ω-categorical and ω-stable, and it follows from the paper of Lachlan that it is coordinatized by indiscernible sets (this even shown for structures which interpret in a dense linear order).
Therefore, we have the following implications:
interprets in countable pure set → ω-categorical, ω-stable, with disintegrated strongly minimal sets → interprets in countable dense linear order.
The second implication cannot be reversed (the dense linear order itself is not ω-stable). My question is whether the first implication can be reversed. In other words, this is the same question as asking about the existence of A totally categorical structure with trivial geometry which is not interpretable in the trivial structure.