The $F$-test for equality of variances needs both independence and normality

Suppose $\{X_1,X_2,\dots,X_m\}$ is a random sample from the distribution of X with mean $\mu_1$ and variance $\sigma_1^2$, $\{y_1,\dots,y_n\}$ is a random sample from the distribution of Y with mean $\mu_2$ and variance $\sigma_2^2$. The test statistic to test $H_0:\sigma_1^2=\sigma_2^2$ against $H_1:\sigma_1^2\neq \sigma_2^2$ is $T_n=S_X^2/S_Y^2$, where $S_X^2$ and $S_Y^2$ are sample variances. Then the null distribution of $T_n$ is: