Minimal condition $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $, $\mid \rho_{ij} \mid \leq 1$, $s_i \in \mathbb R$ and $\Psi_{ij} \in \{0,1\}$

Question

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below

$$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$

$$G = \sum s_i$$

where $\rho = ( \rho_{ij})$ is a correlation matrix with $0 \leq \rho_{ij}<1$ and $\Psi_{ij}= \mathbf 1_{ \{s_i,~s_j < 0 \}^C}$

Assume $\sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \geq 0$ and that there is a $(i,j)$ such that $s_i, s_j < 0$ ($\Psi_{ij} = 0$) and $\rho_{ij} > 0$.

I would like to check if under the above conditions we have that $$F \geq \mid G \mid. $$

Here is the beginin of my reasoning.

Following the conditions above we have the strict inequalities below \begin{align*} F^2= \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j &< \sum ~\rho_{ij}~ s_i ~s_j \\ &< \sum s_i ^+ ~s_j^+ + \sum s_i ^- ~ s_j^- + 2 \sum \rho_{ij} ~s_i ^+ ~ s_j^- \end{align*}

I am stuck at that point. Could someone give me a hand with that?

If it is not possible to show the wanted inequality under those conditions, I would like to know the minimal conditions for it be satisfied.

Any help is appretiate. Thanks