Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sum_{i=1}^nu_i^2v_i^2$, $Y=u_1^2v_1^2$. Consider the following ratio of expectation:
$$r_n(\alpha)=\frac{\mathbb{E}\{\exp[-\alpha^2\frac{1-X}{2}]\}}{\mathbb{E}\{\exp[-\alpha^2(1-Y)]\}}$$
Does there exist a finite upper bound for $r_n(\alpha)$, independent of $\alpha$, for all $\alpha\geq0$? I'm interested in the behavior with fixed $n$ and large $\alpha$.

Update:

I did some simulation, the answer seems to be negative. For any fixed $n$, as $\alpha$ gets large, the ratio seems to diverge to infinity. I have tried $n=2,3,4$ and $\alpha^2$ ranges from 1 to 2000, the log ratio grows linearly. Below is a plot for $n=4$ using Monte Carlo simulation by averaging 20000 samples on the numerator and denominator respectively. 
[![Growth of log ratio][1]][1]


  [1]: https://i.sstatic.net/xJMS0.jpg