Microlocal-sheaf-theoretic proof of local rigidity

Determine whether the local rigidity lemma for Liouville regular Jordan arcs that are $C^1$-embedded on an open subinterval admits a proof using microlocal sheaf theory.

Background

The paper proves a local rigidity lemma stating that if a Liouville regular Jordan arc is $C^1$ on an open subinterval, then every Liouville primitive agrees there, up to endpoint normalization, with the ordinary integral of the Liouville form. This lemma is used in the converse direction of the spiral criterion to show that divergence of the improper signed-area integral prevents Liouville regularity.

The authors note that the converse direction of the spiral theorem can alternatively be established using microlocal sheaf theory, but they leave unresolved whether the local rigidity lemma itself can be proved by such methods. Thus the open problem concerns a microlocal-sheaf-theoretic derivation of the local rigidity statement, not the validity of the lemma, which is already established by a geometric argument.