Howie's conjecture on solvability of non-singular systems over groups

Let $\{w_j=1\}_{j\in J}$ be a system of equations with variables $\{x_i\}_{i\in I}$ over a group $G$. The system is non-singular when the rows of exponent sums of the variables in the words are linearly independent over $\mathbb Q$. A system is solvable over $G$ if $G$ embeds into a group containing a solution of the system. Howie's conjecture. Any non-singular system of equations over any group is solvable over that group. This conjecture generalizes the classical finite-group solvability theorem for non-singular systems; it is stated here as neither proved nor disproved.