  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.
