Nonexistence of infinite finitely generated solvable groups with square-free generalized identity length

Let $G$ be an infinite finitely generated solvable group, and let $n$ be a square-free integer. Nonexistence conjecture. There are no such groups $G$ satisfying a generalized identity of length $n$. The surrounding results construct infinite finitely presented metabelian examples when $p^2\mid n$ for a prime $p$, while the square-free case is left unresolved in the supplied text.