Tresser's period-spectrum conjecture for area-contracting disk embeddings

Let $f$ be a $C^k$ orientation-preserving embedding of the $2$-disk, with $k>1$, and suppose that $f$ is area contracting. Say that $f$ belongs to the boundary of positive topological entropy when it lies on the boundary of the set of maps with positive topological entropy. For a given integer $m$, the periods $m2^j$ for $j\geq 0$ form a period-doubling sequence.

Tresser's conjecture. Generically, maps in the boundary of positive topological entropy have a set of periodic orbits which, except for a finite subset, consists of infinitely many periodic orbits with periods $m2^j$ for a given $m$ and all $j\geq 0$.

The paper states that mildly dissipative diffeomorphisms of the disk provide an affirmative answer in the setting considered there, via a result describing the period set as a finite exceptional family together with finitely many families of the form $m_i2^j$. The conjecture is therefore presented as resolved in that setting, although its full generic formulation concerns the broader space of area-contracting embeddings.