Robust Conditional Variance in the Bivariate Normal

Let $(X, Y)$ follow a bivariate normal distribution with $E[X] = 0$, $E[Y] = 0$, $\operatorname{Var}(X) = 1$, $\operatorname{Var}(Y) = \sigma_Y^2$, and $\operatorname{Corr}(X,Y) = \rho$. Derive $\operatorname{Var}(Y \mid X = x)$ and show that it does not depend on $x$. Evaluate numerically for $\sigma_Y = 3$ and $\rho = 0.6$.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.