Algorithms: lower limit for the longer growing subsequence

I did some research and I could not find an algorithm that would take fewer comparisons than $$n log_ {2} n – n log_ {2} log_ {2} n + O (n)$$ which $$O (n log_ {2} n)$$ in asymptotic terms.

Has it been proven that the lower limit for the problem of the subsequence that increases more in the long term is $$O (n log_ {2} n)$$ in the comparison model? Or maybe it has even been shown to be lower than that?