8 problems

The Primitivity Conjecture. Modulo the agreement of certain orders up to reversal, every primitive homogeneous finite-dimensional permutation structure is the Fraïssé limit of all…

Construction completeness conjecture. This construction produces all homogeneous finite-dimensional permutation structures.

Homogeneity-after-naming conjecture. The class $C_2$ remains homogeneous after naming $C_1$.

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

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

Classification conjecture. Every homogeneous finite-dimensional permutation structure with lattice of $\emptyset$-definable equivalence relations isomorphic to $\Lambda$ is interde…

Genericity conjecture. Then $93$ is fully generic.

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