The Inscribed Affine-Regular Hexagon Problem

Question: does every centrally symmetric convex body K ⊂ ℝ² contain an inscribed affine-regular hexagon (i.e., a hexagon that maps to a regular hexagon under a linear transformation)?

Related to the square-peg problem (also on our shelf) via the affine geometry pipeline: if true for all centrally symmetric bodies, the general case follows by symmetrisation. Partial results exist for smooth bodies (make it a consequence of Borsuk–Ulam type arguments). The hexagon case is the first genuinely open step beyond squares in the inscribed-polygons hierarchy.