Jordan property for Cremona groups

Assuming the Borisov-Alexeev-Borisov conjecture, we prove that there is a constant $J=J(n)$ such that for any rationally connected variety X of dimension n and any finite subgroup G⊂ Bir(X) there exists a normal abelian subgroup A⊂ G of index at most J. In particular, we obtain that the Cremona group Cr₃= Bir( P³) enjoys the Jordan property.