Finite Hirsch length and nuclear dimension for elementary amenable groups

Let $G$ be a finitely generated elementary amenable group. Its Hirsch length is the dimension-like invariant obtained from the ranks of the abelian factors in an appropriate elementary amenable series, and $C^*(G)$ denotes its group C*-algebra.

Finite Hirsch length conjecture. The group $G$ has finite Hirsch length if and only if

$$C^*(G)\text{ has finite nuclear dimension}.$$

The main theorem establishes finite nuclear dimension for group C*-algebras of virtually polycyclic groups, and the known examples of elementary amenable groups with infinite nuclear dimension have infinite Hirsch length. The converse and the full equivalence remain open for finitely generated elementary amenable groups.