Salamon-Walpuski self-wedge conjecture

Prove that every 4-form α on an 8-dimensional real vector space satisfying α∧α=0 is degenerate in the Salamon-Walpuski sense: for some linearly independent triple u,v,w, no vector x satisfies α(u,v,w,x)≠0.

Background

The paper adopts Salamon and Walpuski’s definition of non-degeneracy: a 4-form is non-degenerate if every linearly independent triple of vectors can be extended by some fourth vector on which the form is nonzero. This condition is stronger than multisymplectic non-degeneracy.

The central unresolved conjecture asserts that vanishing self-wedge forces degeneracy. The paper proves related characterizations of non-degeneracy and rules out strongly non-degenerate forms with vanishing self-wedge, but does not prove the full implication.

References

Let \alpha be a 4-form on V. If \alpha\wedge\alpha=0, then \alpha is degenerate.

— A note on 4-forms in 8-dimensions  (2608.20200 - Close, 20 Aug 2026) in Section 3, Conjecture (Salamon-Walpuski), following Definition 3.1

What is not clear, and is the reason why we cannot answer the conjecture, is whether or not strong non-degeneracy is equivalent to non-degeneracy, that is, \begin{equation} \text{strong non-degeneracy}\iff \text{non-degeneracy?} \end{equation}

— A note on 4-forms in 8-dimensions  (2608.20200 - Close, 20 Aug 2026) in Section 3, concluding discussion before the final paragraph on metric 4-forms