Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence. Furthermore, let $k=\omega(1)$ increase, and $n=\omega(k)$ increasing faster than $k$ but not too much fast, i.e. $\log(n)=o(t)$. For example, $n=k^2$. This is the same scenario as in my previous question.

Now, let

$$S(n)=\frac{1}{n}\sum_{i=1}^n\exp\left[\frac{\sqrt{\log n}}{\sqrt{k}}(X_i-X_\max)\right]$$

I am wondering about the limiting behavior of $S(n)$ as $n\rightarrow\infty$. My conjecture (based on the fact that the maximum isn't too far away from the rest of the sequence of random variables with exponential tail, as well as the answer and comment to the two related questions) is that $S(n)$ converges to unity in distribution, however, I am having hard time proving this formally, as I get bogged down in a nasty convolution when I try to do it (but, maybe I am approaching this wrong and perhaps I am incorrect). Any suggestions?