Regression beats both, but mean versus ratio depends on $\rho$

Let us consider the following methods of estimation, all based on SRSWOR of the same sample size. I Sample mean, II Ratio estimator, III Regression estimator. Let the corresponding variances be denoted by $V_1^2$, $V_2^2$ and $V_3^2$. Assuming that the sample size is sufficiently large, we have: