Universal frame set for rational functions

Let $g\in L^2(\mathbb{R})$ be a rational function of degree $M$, i.e. there exist polynomials $P,Q$ such that $g=\frac{P}{Q}$ and $deg(P)<deg(Q)\le M$. We prove that for any $\epsilon >0$ and any $M\in \mathbb{N}$ there exists universal set $\mathrm{\Lambda }\subset \mathbb{R}$ of density less than $1+\epsilon$ such that the system $$\{e^{2\pi i\lambda t}g(t-n)\text{\hspace{0.17em}⁣}:(\lambda ,n)\in \mathrm{\Lambda }\times \mathbb{Z}\}$$ is a frame in $L^2(\mathbb{R})$ for any well-behaved rational function $g$.