Quadratic Schreier growth gap conjecture for virtually torsion-free solvable groups

Let $G$ be a finitely generated solvable group which is virtually torsion-free and not virtually abelian. A Schreier growth gap $n^2$ means that the relevant Schreier growth of $G$ has a quadratic lower bound.

Quadratic Schreier growth gap conjecture. $G$ has a Schreier growth gap $n^2$.

This conjecture asks whether the quadratic Schreier growth gap known for torsion-free metabelian groups extends to virtually torsion-free solvable groups. The source does not provide a resolution.