String C-groups as transitive subgroups of SnSn

If Γ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group SnSn (other than SnSn and the alternating group AnAn), then the rank of Γ is at most n/2+1n/2+1, with finitely many exceptions (which are classified). It is conjectured that only the symmetric group has to be excluded.