Homogeneity of equivalence classes after naming another class

Let $E$ be a $\emptyset$-definable equivalence relation in a finite-dimensional permutation structure, and let $C_1$ and $C_2$ be $E$-classes. Naming $C_1$ means expanding the structure by a predicate for $C_1$.

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

The statement is identified as a possible route to ruling out counterexamples to the conjectural classification and is known in the case of three linear orders; the general claim remains open.