Does correlation of inverse covariance matrix implies conditional independence?

Question

I have x $= (x_1,\dots,x_n)^T \in \mathbb{R}^n$ such that x $\sim \mathcal{N_n}\left(\mu, \Sigma_x\right)$ $\Sigma_x=\Sigma_x^T$ and is positive definite matrix, can i proof that $(\Sigma_x^{-1})_{ij}=0, i\neq j \implies x_i ⫫ x_j | \left\{x_k : k \neq i, k \neq j\right\}$ ? I get joint distributions with Brook's Lemma but i couldn't get any advance from there