Consider a covariance function (positive definite function) on $\mathbb{Z}$: $$ \gamma(k)=(1+|k|)^{-\alpha},\quad \alpha>0. $$ It is guaranteed to be positive definite by Polya's criterion (monotonicity and convexity). Consider matrices $$ \Sigma_1=[\gamma(i-j)]_{1\le i,j\le m},\quad \Sigma_2=[\gamma(k+i-j)]_{1\le i,j\le m},\quad m>0,~k\ge 0. $$ Show that all entries of $\Sigma_1^{-1}\Sigma_2$ are nonnegative.

Remark: I checked it numerically for different $k$, $m$ and $\alpha$. A further question is to identify general conditions on a positive definite $\gamma(k)$ which fulfills the preceding requirement.