Strong Nielsen equivalence on the punctured disc

Let $f$ be an orientation-preserving homeomorphism of the 2-disc $\mathbb{D}^2$ that fixes the boundary pointwise and leaves invariant a finite subset in the interior of $\mathbb{D}^2$. We study the strong Nielsen equivalence of periodic points of such homeomorphisms $f$ and we give a necessary and sufficient condition for two periodic points to be strong Nielsen equivalent in the context of braid theory. In addition, we present an application of our result to the trace formula given by Jiang--Zheng, deducing that the obtained forced periodic orbits belong to different strong Nielsen classes.