Rational certificates of non-negativity on semialgebraic subsets of cylinders

Let $g_1,\dots ,g_s\in \mathbb{R}[X_1,\dots ,X_n,Y]$ and $S=\{(\bar{x},y)\in {\mathbb{R}}^{n+1}\mid g_1(\bar{x},y)\ge 0,\dots ,g_s(\bar{x},y)\ge 0\}$ be a non-empty, possibly unbounded, subset of a cylinder in ${\mathbb{R}}^{n+1}$. Let $f\in \mathbb{R}[X_1,\dots ,X_n,Y]$ be a polynomial which is positive on $S$. We prove that, under certain additional assumptions, for any non-constant polynomial $q\in \mathbb{R}[Y]$ which is positive on $\mathbb{R}$, there is a certificate of the non-negativity of $f$ on $S$ given by a rational function having as numerator a polynomial in the quadratic module generated by $g_1,\dots ,g_s$ and as denominator a power of $q$.