Distinguishing even-dimensional loose Legendrian classes

Construct a method to distinguish the two classes of loose Legendrians with fixed Thurston–Bennequin invariant and rotation class when the Legendrian dimension is even.

Background

The paper notes that, unlike the odd-dimensional case, even-dimensional loose Legendrians can have two distinct formal Legendrian isotopy classes for the same Thurston–Bennequin invariant and rotation class. The authors do not know an invariant or procedure that distinguishes these classes.

References

As commented in Appendix, we do not know how to distinguish these two classes.

— Floor-crossing for Legendrian Clasp Move  (2609.01381 - Strakoš, 1 Sep 2026) in Remark following Proposition 1.4, Section 2.1 (Legendrian Unknotting Number and Loose Legendrians)