3-type realization conjecture for homogeneous finite-dimensional permutation structures

Let $\Gamma$ be a homogeneous finite-dimensional permutation structure, and call it fully generic when it is the generic structure obtained from the relevant amalgamation class.

3-type realization conjecture. If $\Gamma$ realizes all 3-types, then $\Gamma$ is fully generic.

This is presented as one part of the Primitivity Conjecture and is retained for consideration via direct amalgamation arguments; the source gives no resolution.