Higher transversality conjecture for the involution $\omega$

Let $R$ be the ring under consideration, let $S$ be its $\omega$-invariant subring, and let $I$ be the relevant ideal. An involution is $n$-transverse when it satisfies the $n$-transversality condition defined in the paper.

Higher transversality conjecture. The involution $\omega$ is also $n$-transverse for every integer $n\ge 3$.

The paper has established the First and Second Transversality Theorems, and proposes this higher-transversality assertion as an open extension of those results.