 Free conjugate bases in a rank-two free group

In $F(a,b)$, any set of distinct conjugates $b^{-i}a{b}^i$ freely generates its subgroup. Combine adjacent powers of the same conjugate in a reduced word. After expansion, distinct successive indices leave nonzero intervening powers of $b$, and the intervening nonzero powers of $a$ cannot cancel in the free product $⟨a⟩\ast ⟨b⟩$. The same argument works with $a,b$ interchanged and for finite or countably infinite sets of indices.