Openness of the frame set on the hyperbolas

We prove that for the functions of the form $g(x)=h(x)+\frac{C}{x+i}$, where $h$ belongs to the continuous Wiener algebra $W_0$, the intersection of the frame set ${\mathcal{F}}_g$ with every hyperbola $\{\alpha ,\beta >0\mid \alpha \beta =c\}$ is open in the relative topology. In particular, this applies to all rational functions $g$.