Teheranchi's strong Gaussian correlation conjecture for Gaussian rectangles

Let $X_1,\dots, X_n$ be zero-mean jointly Gaussian real random variables, and let $s_1,\dots, s_n, t_1,\dots, t_n \in [0,\infty)\cup \{\infty\}$. Teheranchi's strong Gaussian correlation conjecture.

$$\begin{aligned} \Pr(|X_i|\le s_i+t_i \ \forall i\in [n])&\Pr(|X_i|\le \min\{s_i,t_i\} \ \forall i\in [n])\\ &\ge \Pr(|X_i|\le s_i \ \forall i\in [n])\Pr(|X_i|\le t_i \ \forall i\in [n]). \end{aligned}$$

This conjecture strengthens the Gaussian correlation inequality and is equivalent to a conjecture of Teheranchi. The paper proves special cases and shows that an asymptotic version would imply the conjecture in every dimension; the general statement remains open.