Sela–Kharlampovich–Myasnikov characterization of groups elementarily equivalent to free groups

Let $G$ be a finitely generated group. A fully residually free tower is a group obtained from finitely generated free groups and surface groups by taking free products, free extensions of centralizers, and gluing a surface group onto a base group so that the surface group retracts onto the base group. A group is non-elementary hyperbolic if it is Gromov-hyperbolic and not elementary.

Characterization conjecture. The group $G$ is elementarily equivalent to a non-abelian free group if and only if $G$ is a non-elementary hyperbolic fully residually free tower.

This is a proposed characterization arising from the announced positive solution of Tarski's problem. The source states it as a conjecture because the referring processes for the announced results were not yet completed.