Howie's conjecture: every nonsingular system of equations over a group is solvable in an overgroup

Claims Theorem 1: for every group $G$ and every finite system $w_1,\dots,w_m\in G*F_n$ whose exponent-sum matrix has rank $m$ over $\mathbb Q$, the canonical map $G\to(G*F_n)/\langle\langle w_1,\dots,w_m\rangle\rangle$ is injective, so the system has a simultaneous solution in an overgroup. This contains the Kervaire-Laudenbach conjecture (one equation, one unknown, nonzero exponent sum) and the Kervaire conjecture. It does NOT control the overgroup (no finiteness, solvability or other structure of the solution group), and says nothing about singular systems.

Claims Theorem 1: for every group $G$ and every finite system $w_1,\dots,w_m\in G*F_n$ whose exponent-sum matrix has rank $m$ over $\mathbb Q$, the canonical map $G\to(G*F_n)/\langle\langle w_1,\dots,w_m\rangle\rangle$ is injective, so the system has a simultaneous solution in an overgroup. This contains the Kervaire-Laudenbach conjecture (one equation, one unknown, nonzero exponent sum) and the Kervaire conjecture. It does NOT control the overgroup (no finiteness, solvability or other structure of the solution group), and says nothing about singular systems.