It is very good how many people interested in this conjecture. Sorry for not taking part in your discussion so far, but I am trying to find more information about harmonic numbers which can be useful in the investigation of this problem. Hope cooperative work bring results.

There is a question (link in the discription) to prove that there exist infinitely many $n$ such that $$H_n - \lfloor H_n\rfloor < \frac{1}{n^2}.$$ So I wanted to find several $n$'s numerically, but I did not find any $n > 1$ which fulfilled that inequality. Also, see pictures I added in the description.

Yes. It denotes cardinality,

I got it, Like a tag question.

I agree that $\bar\eta$ is decreasing. But I do not understand the ''isn't'' in the beginning.