An isomorphism of the free group factors

We solve the free group factor isomorphism problem affirmatively by proving that $L({\mathbb{F}}_2)$ and $L({\mathbb{F}}_3)$ are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including $L({\mathbb{F}}_{\infty })$, are isomorphic and have fundamental group ${\mathbb{R}}_{>0}$.