0-GAS and bounded input–frequently bounded state imply iISS

Consider the time-delay system on $\mathcal X^n$ with inputs in $\mathcal U^m$. 0-GAS plus BEFBS conjecture. Assume the system is 0-GAS and, for some $\zeta\in\mathcal K_\infty$, every $x_0\in\mathcal X^n$ and $u\in\mathcal U^m$ satisfy

$$\int_0^{+\infty}\zeta(|u(\tau)|)\,d\tau<+\infty\quad\Longrightarrow\quad\liminf_{t\to+\infty}|x(t,x_0,u)|<+\infty.$$

Then the system is iISS. The implication implicitly includes forward completeness; the claim is presented as the time-delay analogue of a result known for delay-free systems, and its resolution is not supplied.