  We prove Turner's conjecture, which describes the blocks of the Hecke
algebras of the symmetric groups up to derived equivalence as certain explicit
Turner double algebras. Turner doubles are Schur-algebra-like `local' objects,
which replace wreath products of Brauer tree algebras in the context of the
Brou\'e abelian defect group conjecture for blocks of symmetric groups with
non-abelian defect groups. The main tools used in the proof are generalized
Schur algebras corresponding to wreath products of zigzag algebras and
imaginary semicuspidal quotients of affine KLR algebras.
