Centrally symmetric strong isotropic constant conjecture (parallelepiped maximizes)

Prove that for every centrally symmetric convex body K ⊂ R^n, the isotropic constant satisfies L_K ≤ L_{C_n} = 1/√12, where C_n denotes an n-dimensional parallelepiped. This would identify parallelepipeds (affine images of cubes) as global maximizers among centrally symmetric convex bodies.

Background

The paper records a symmetric analogue of the strong conjecture, positing that among centrally symmetric convex bodies, the maximum isotropic constant is achieved by parallelepipeds with value 1/√12, independent of dimension.

The authors provide further structural evidence by showing that, under local maximizing assumptions with certain boundary regularities (e.g., a simplicial vertex or a cubical zone in a zonotope), the extremal body must be a cross-polytope or a cube, respectively—supporting the conjectured form of maximizers.