Kervaire–Laudenbach–Howie conjecture on independent systems of equations

Let $G$ be a group. An independent system of equations over $G$ is a finite system of equations whose exponent-sum matrix has full row rank. Such a system is said to have a solution in an overgroup of $G$ if the natural map from $G$ into the corresponding quotient of the free product is injective.

Kervaire–Laudenbach–Howie conjecture. Every independent system of equations over $G$ has a solution in an overgroup of $G$.

This is one of the most important open problems in the theory of equations over groups.