Let $B_i(n,1/2)$ be independent identically distributed binomial random variables. I am interested in the asymptotic growth of the maximum of $n$ such random variables. In https://math.stackexchange.com/questions/590880/bounds-for-the-maximum-of-binomial-random-variables the asymptotics were shown to be bounded above by $\frac{n}{2}+ \sqrt{\frac{n}{2} \log_e n}$ and numerically shown to be bounded below by $\frac{n}{2}+ \sqrt{\frac{n}{2.5} \log_e n}$.

Is it possible to derive the exact asymptotic behaviour for the maximum of $n$ such independent and identical $B_i(n,1/2)$ binomial random variables?

It seems (very) likely this is a solved problem but I haven't found where or by whom yet. I would be happy with a reference if that is available.