GAS versus Lyapunov stability and global attractivity for time-delay systems

Let the autonomous time-delay system be given on the history space $\mathcal X^n$, with solution $x(t,x_0)$ from initial history $x_0$. GAS conjecture. The system is GAS if and only if it is stable, meaning that for every $\varepsilon>0$ there exists $\delta>0$ such that $\|x_0\|\leq\delta$ implies $|x(t,x_0)|\leq\varepsilon$ for all $t\geq0$, and globally attractive, meaning that $x(t,x_0)\to0$ as $t\to+\infty$ for every $x_0\in\mathcal X^n$. This asks whether the uniformity implicit in GAS follows for time-delay systems from stability and global attractivity without assuming RFC; the parser reports that the conjecture is disproved.