Covariance of sigmoid function of normal random variables

Question

$\newcommand{\Cov}{\operatorname{Cov}}$Suppose $X=(X_1,X_2)^T \sim N(\mu, \Sigma)$ and $p=(p_1,p_2)^T = (e^{X_1}/(1+e^{X_1}),e^{X_2}/(1+e^{X_2}))^T$. $\Cov(X_1,X_2)=\sigma_{12}$. Does the following hold ? I guess it is right, but I can not prove this.

1. $\sigma_{12}=0 \Leftrightarrow \Cov(p_1,p_2)=0$
2. $\sigma_{12} > 0 \Leftrightarrow \Cov(p_1,p_2) > 0$
3. $\sigma_{12} < 0 \Leftrightarrow \Cov(p_1,p_2) < 0$

Edit I only have the result that two uncorrelated normal random variables are independent, hence $\sigma_{12}=0 \Rightarrow \Cov(p_1,p_2)=0$. I did several simulations and found no counter examples.