3-type extension conjecture for primitive homogeneous finite-dimensional permutation structures

Let $\Gamma$ be a primitive homogeneous finite-dimensional permutation structure. A realized 2-type is a 2-type realized in $\Gamma$, and a 3-type involves such a 2-type when its restriction to a pair is realized in $\Gamma$.

3-type extension conjecture. All 3-types involving realized 2-types are realized in $\Gamma$.

This is presented as the second part of the Primitivity Conjecture; the source gives no resolution.