On lattices whose minimal vectors form a 6-design

Let L be a lattice of dimension n≤24 such that the minimal vectors of L form a 6-design and generate L. Then L is similar to either the root lattice E8, the Barnes-Wall lattice BW16, the Leech lattice Λ24, or n=23. For n=23 we conjecture that the only possibilities for L are the shorter Leech lattice O23 or its even sublattice Λ23.