Strengthened Hanna Neumann conjecture

Statement

If H, K ≤ G are two subgroups of a group G and if a, b ∈ G define the same double coset HaK = HbK then the subgroups H ∩ aKa⁻¹ and H ∩ bKb⁻¹ are conjugate in G and thus have the same rank. It is known that if H, K ≤ F(X) are finitely generated subgroups of a finitely generated free group F(X) then there exist at most finitely many double coset classes HaK in F(X) such that H ∩ aKa⁻¹ ≠ {1}. Suppose that at least one such double coset exists and let a₁,...,a_n be all the distinct representatives of such double cosets. The strengthened Hanna Neumann conjecture, formulated by her son Walter Neumann (1990), states that in this situation

$$\sum_{i=1}^n [{\rm rank}(H\cap a_iKa_{i}^{-1})-1] \le ({\rm rank}(H)-1)({\rm rank}(K)-1).$$

The strengthened Hanna Neumann conjecture was proved in 2011 by Joel Friedman. Shortly after, another proof was given by Igor Mineyev.