Finite move sets yielding monotonic simplification

Construct a natural finite collection of local petal-number-preserving moves, beyond crossing exchanges, with respect to which every petal diagram of the unknot admits monotonic simplification, subject to the constraint that the moves cannot all be composites passing through only two additional petals.

Background

The allowed move set M consists of trivial petal additions and deletions, crossing exchanges, and two petal-number-preserving symmetries. The paper exhibits hard 9-petal unknot diagrams under this move set, showing that monotonic simplification fails. It also proves that the hardest example cannot be simplified by augmenting the move set with composites that pass through diagrams having at most two additional petals. The open problem asks whether a broader but still finite and natural collection of local, petal-number-preserving moves can restore a Dynnikov-type monotonic simplification theorem.