Suppose $(Z_1, Z_2)$ is the zero-mean bivariate normal distribution with covariance $\left( \begin{matrix} 1 & \rho; \\ \rho & 1\end{matrix} \right)$ with positive $\rho > 0$. What I want to know a valuable tighy lower bound of the probability $$\mathrm{P} \left( Z_1 > z_1, Z_2 < z_2\right)$$ where $z_1, z_2$ are some positive constants.

PS: There are various valuable lower bound for the tail probability $\mathrm{P} \left( Z_1 > z_1, Z_2 > z_2\right)$ with positive $\rho > 0$, see p495-p499 in Continuous Bivariate Distributions. Can we transform $\mathrm{P} \left( Z_1 > z_1, - Z_2 > -z_2\right)$ to $\mathrm{P} \left( Z_1 > z_1, Z_2 > z_2\right)$ keeping the correlation coefficient positive?