The universal solvability conjecture for unimodular equations

An equation $w(x)=1$ over a group $G$, where $w(x)\in G\ast\langle x\rangle_\infty$, is unimodular if the exponent sum of $x$ in $w(x)$ equals $\pm1$. It is solvable if there exists a group $\widetilde G$ containing $G$ and an element $\widetilde g\in\widetilde G$ such that $w(\widetilde g)=1$. Universal solvability conjecture. Any unimodular equation over any group is solvable. The conjecture extends known solvability results for finite, residually finite, locally indicable, hyperlinear, and torsion-free groups; its general case remains open.