Classification conjecture for homogeneous finite-dimensional permutation structures

Let $\Lambda$ be a lattice, and let a homogeneous finite-dimensional permutation structure mean a homogeneous structure presented by finitely many linear orders, with its lattice of $\emptyset$-definable equivalence relations isomorphic to $\Lambda$. Let a well-equipped lift be the class construction described in the source, and let $\Lambda$-ultrametric spaces be finite structures equipped with the corresponding lattice-valued ultrametric.

Classification conjecture. Every homogeneous finite-dimensional permutation structure with lattice of $\emptyset$-definable equivalence relations isomorphic to $\Lambda$ is interdefinable with the Fraïssé limit of some well-equipped lift of the class of all finite $\Lambda$-ultrametric spaces.

This is the thesis's proposed classification of all homogeneous finite-dimensional permutation structures; the source gives no resolution of the conjecture.