The primitive intermediate-growth conjecture for the circular order

Let $c9$-categorical mean categorical in every infinite cardinality, and for an $c9$-categorical structure $c4$ let $u_n(c4)$ denote the number of orbits of $c0 goperatorname{Aut}(c4)$ on $n$-element subsets of $c4$. A structure is bidefinable with another when each is definable in the other. Let $S(2)$ be the generic two-tournament structure and let its betweenness reduct be the reduct mentioned in the source.

Primitive intermediate-growth conjecture. If a primitive $c9$-categorical structure $c4$ satisfies

$$\frac{2^n}{p(n)}<u_n(c4)<\frac{2^n}{q(n)}$$

for some polynomials $p$ and $q$, then $c4$ is bidefinable with $S(2)$ or its betweenness reduct.

The conjecture proposes a classification of primitive $c9$-categorical structures with orbit growth lying between two polynomially corrected copies of $2^n$. The source presents it after noting that the circular-order betweenness structure has the slowest known primitive growth in this range; it remains open.