What is conditional normal variance of $X$ given that dependent $Y$ is bigger than a real number $M$? [duplicate]

There are two dependent normal variables with the same distribution and the correlation coefficient $$\rho$$: $$X,Y \sim N(\mu, \sigma^2)$$.

I would like to get $$P(X>M|Y>M)$$. For that I need to know the conditional expectation and variance. But I can not find any information about the conditional normal variance in the case when $$Y$$ is smaller/bigger than some real number.

There is not a comparison between two normal variables. And it's not the truncated normal distribution.

I found the conditional expectation of $$X$$ given that $$Y$$ is bigger than $$M$$ is $$E(X|Y>M)= \mu + \rho \sigma \frac{\phi(\frac{M-\mu}{\sigma})}{1-\Phi(\frac{M-\mu}{\sigma})}$$.

But what is the conditional variance $$var(X|Y>M)$$? Is it $$(1-\rho^2)\sigma^2$$, as it would be in the case of $$var(X|Y=M)$$, where variance does not depend on $$M$$?