In a survey of a population of size $N=2000$, a researcher uses a two-phase (double) sampling design to estimate the mean $\bar{Y}$. The first phase, with sample size $n'=500$, is used to estimate the mean of an auxiliary variable $\bar{X}$. The second phase, with $n=100$, measures $Y$. If the regression estimator is defined as $\bar{y}_{lr} = \bar{y} + b(\bar{X}' - \bar{x})$, where $b$ is the sample regression coefficient, which of the following statements about the variance of $\bar{y}_{lr}$ are true?