The totally decomposable word conjecture

Let $w$ be a finite word in the alphabet $\{X,A\}$. A word is totally decomposable if it is the image of $X$ under a composition of maps $\pi_{m,k}:u\mapsto (uA^k)^m u$ for $m\geq 1$, $k\geq 0$, together with $r:u\mapsto uA$ and $l:u\mapsto Au$. A word is universal if, for every uniquely divisible group $G$ and every $A,B\in G$, the equation $w(X,A)=B$ has a solution $X\in G$; it is uniquely universal if that solution is always unique. A solution is in terms of radicals when it is obtained by the radical operations defined for the free uniquely divisible group in the paper.

The totally decomposable word conjecture. The following are equivalent: $w$ is totally decomposable; $w$ is uniquely universal; $w$ is universal; and $w(X,A)=B$ has a solution in terms of radicals.

The implications from total decomposability to unique universality and from unique universality to universality, as well as the equivalence with radical solvability, are straightforward. The difficult implication is that radical solvability forces total decomposability; the authors verified the conjecture computationally for words of length at most $10$.