Visibility under iterations in local symplectic/Floer homology

Ascertain whether a closed Reeb orbit can become F-visible under iteration when the original prime closed orbit is F-invisible, without additional assumptions on the orbit, the ground field, or the iteration order k.

Background

Local symplectic and Floer homology detect homologically visible closed orbits. The paper discusses the complex and poorly understood behavior of visibility under iteration. While examples show that an F-visible orbit can become invisible under certain iterations, the converse (invisible becoming visible under iterations) is unknown in general.

This question is central to understanding how periodic orbit data changes under iteration and impacts global filtered and local homology structures.