How to prove the probability $P(Z_1 > w, |Z_2| < \theta)$ achieves its maximum when they are independent

Question

everyone. I am currently stuck on a proof and I need some help. I am trying to prove that $P(Z_3 > w, |Z_2| < \theta) > P(Z_1 > w, |Z_2| < \theta)$, where $Z_1, Z_2$, and $Z_3$ are standard normal distributions. $Z_3$ and $Z_2$ are independent, while $Z_1$ and $Z_2$ are correlated. The values of $w$ and $\theta$ can be any positive number. In other words, I want to prove that the probability reaches its maximum when the correlation coefficient is equal to $0$. This proof is pretty important to me. Thank you in advance for the help.