Maximum of sub-gaussian 3

Question

Let $X_1, X_2,\cdots$ be an infinite sequence of sub-gaussian random variables which are not necessarily independent. Show that \begin{align*} \mathbb {E}\text{max}_{i}\frac {|X_i|}{\sqrt{1+\log i}}\leq CK, \end{align*} where $K=\text{max}_{i}\|X_i\|_{\psi_{2}}$. Deduce that for every $N\geq 2$ we have \begin{align*} \mathbb {E}\text{max}_{i\leq N}|X_i|\leq CK \sqrt{\log N}. \end{align*} This question is the Exercise 2.5.10 in https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf, and has been asked twice, you can find them here:Expected maximum of sub-Gaussian and Upper bound of expected maximum of weighted sub-gaussian r.v.s. But I don't think the answer is satisfactory since it did not give $CK$ explicitly.