We prove that for every $n\in \mathbb{N}$ such that $n\geq 2$ , the reduced group $C^{*}$ -algebras of the countable free groups $C^{*}_{r}(\mathbb{F}_{n})$ have strict comparison. Our method works in a general setting: for every finitely generated acylindrically hyperbolic group $G$ with trivial finite radical and the rapid decay property, we have $C^{*}_{r}(G)$ have strict comparison. This work also has several applications in the theory of $C^{*}$ -algebras including: resolving Leonel Robert’s selflessness problem for $C^{*}_{r}(G)$ ; uniqueness of embeddings of the Jiang-Su algebra $\mathcal{Z}$ up to approximate unitary equivalence into $C^{*}_{r}(G)$ ; full computations of the Cuntz semigroup of $C^{*}_{r}(G)$ and future directions in the $C^{*}$ -classification program.
