Statistical Inference — equivalence of equal-variance testing and an independence check

Suppose (X, Y) follows bivariate normal distribution with means $\mu_1,\mu_2$; standard deviations $\sigma_1,\sigma_2$ and correlation coefficient $\rho$, $-1<\rho<1$, where all the parameters are unknown. Then for checking $\sigma_1=\sigma_2$ is equivalent to verifying the independence of