Suppose $X_{\max}$ is the maximum in a sequence $X_1,X_2,\ldots,X_n$ where each $X_i\sim\chi^2_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom.

Since chi squared distribution has an exponential tail, for some fixed number of degrees of freedom $k$ we know that $\lim_{n\rightarrow\infty}\frac{X_{\max}}{\ln n}=c$ almost surely, with $c$ being a constant (see Example 3.5.6 on page 176 here). A weaker convergence in probability claim can be made using the convergence of the distribution of appropriately centered $X_{\max}$ to Gumbel (see summary in Table 3.4.4 on page 156 of the same reference).

However, what happens if both the number of degrees of freedom $k$ also increases, though at a rate that is slower than the increase in the number of random variables in the sequence $n$? For example, let $n=k^2$ (in general, my $n=\omega(k)$). Is there a result similar to the above, with $k\rightarrow\infty$ (which implies that $n\rightarrow\infty$)?

Does convergence of the distribution of $X_{\max}$ to Gumbel even apply with centering in the above-referenced Table 3.4.4 under these conditions? (the norming constant $d_n$ is negative when $n=k^2$ is substituted, so I think there is trouble here.)