Holder continuity of interfaces for scale-invariant Poisson stick soup

We study the interface of covered and vacant sets in the subcritical phase of a scale-invariant Poisson stick soup on the plane. This model is a natural candidate for scaling limit of some planar models and has connections with long-range percolation on the plane with critical parameter $s=4$. We analyze a family of exploration paths on boxes and prove tightness for this family and Holder continuity for its limiting measures.