Suppose $X$ and $Y$ are correlated with correlation coefficient $\rho$. They are jointly normal with means $\mu_X$ and $\mu_Y$ respectively. Then what is $Var[X | Y \geq T]$? Feel free to add additional assumptions if necessary. $E[X | Y \geq T]$ is already computed here: Expectation of a normal random variable when conditioning on a correlated normal random variable being above a threshold, so I just need $E[X^2 | Y \geq T]$.
(Update) Ah, I think this answer is quite relevant, although I still need to tweak it for the case above: https://math.stackexchange.com/questions/3403395/variance-of-truncated-bivariate-normal
