Divisible Rigid Groups. III. Homogeneity and Quantifier Elimination

A group $G$ is said to be rigid if it contains a normal series
$$ G=G_1>G_2>\dots>G_m>G_{m+1}=1, $$
whose quotients $G_i/G_{i+1}$ are Abelian and, treated as right $\mathbb{Z}[G/G_i]$-modules, are torsion-free. A rigid group $G$ is divisible if elements of the quotient $G_i/G_{i+1}$ are divisible by nonzero elements of the ring $\mathbb{Z}[G/G_i]$. Every rigid group is embedded in a divisible one.
THEOREM. Let $G$ be a divisible rigid group. Then the coincedence of $\exists$-types of same-length tuples of elements of the group $G$ implies that these tuples are conjugate via an authomorphism of $G$.
As corollaries we state that divisible rigid groups are strongly $\aleph_0$-homogeneous and that the theory of divisible $m$-rigid groups admits quantifier elimination down to a Boolean combination of $\exists$-formulas.