In the article Chaotic Functions with Zero Topological Entropy by Jaroslav Smítal, in the Remark 2.6 the autor cite the following result,
If $f\in C^0 (I,I)$ has zero topological entropy, and the set of periodic points is closed and nowhere dense, then for any given $\varepsilon>0$, every $g\in C^0 (I,I)$ sufficiently near to $f$ has only trajectories $\varepsilon$-approximable by cycles.
And then conjecture that the result could be valid if the periodic points form a nowhere dense set, that is, it is not necessary for this set to be closed.
This is a 1986 article, and I would like to know if there has been any progress on this conjecture, even some discussion about it, I researched and read some articles and found nothing about it, would anyone have any knowledge about the situation of this result?
Edit:
Following the author's notation (in other articles), it is possible to rewrite the statement he makes as,
If $f\in C^0(I,I)$ has zero topological entropy and the set of periodic points is nowhere dense, then $f$ is stable.
Remembering that $f$ is nonchaotic and for reference to the definition of stable see Definition 2.1.