Genericity conjecture for homogeneous permutation structures realizing all 3-types

Let $93$ be a homogeneous $k$-dimensional permutation structure that realizes all 3-types. A structure is fully generic when it is the Fra39ss39 limit of the relevant amalgamation class of finite structures.

Genericity conjecture. Then $93$ is fully generic.

The paper describes this as a consequence of the Primitivity Conjecture and notes that the case $k=3$ is proved, while the claim appears tractable for several further small values of $k$.