Characterization of the Star Condition

Characterize whether every anticommuting unit family of arbitrary bivectors satisfying the star condition is, up to isomorphism or a basis change, of the form B_i = C S_i with one common generator C and pairwise anticommuting leaf generators S_i.

Background

The paper defines a star family by bivectors B_i = C S_i generated from one common central generator and pairwise anticommuting leaf generators. Every such star family satisfies the star condition, namely that the scalar part of every nonempty subset product vanishes. For basis bivectors, the paper states that the star condition is equivalent to having star rather than triangle structure. The unresolved issue is whether this equivalence extends to arbitrary bivectors, possibly after an isomorphism or change of basis. The authors formulate the affirmative answer as a conjecture but do not prove it.