Braun's Primitivity Conjecture for homogeneous finite-dimensional permutation structures

A primitive homogeneous finite-dimensional permutation structure is a homogeneous finite-dimensional permutation structure with no nontrivial $\emptyset$-definable equivalence relation. The construction in question proceeds by identifying certain orders, up to reversal, and then taking the Fra39ss39 limit of the resulting amalgamation class.

Braun's Primitivity Conjecture. Every primitive homogeneous finite-dimensional permutation structure can be constructed by this procedure, yielding a fully generic structure, possibly in a simpler language.

This classification was conjectured by Braun. The paper shows that it would follow from a positive answer to the broader classification question for homogeneous finite-dimensional permutation structures with distributive lattices of $\emptyset$-definable equivalence relations.