Point-wise integral-to-integral estimate implies ISS for time-delay systems

Let the time-delay system act on $\mathcal X^n$ with inputs in $\mathcal U^m$. Assume that there are $\alpha,\overline\alpha\in\mathcal K_\infty$ and $\gamma\in\mathcal N$ such that, for every initial history $x_0\in\mathcal X^n$, input $u\in\mathcal U^m$, and $t\geq0$, the point-wise integral estimate

$$\int_0^t\alpha(|x(\tau,x_0,u)|)\,d\tau\leq\overline\alpha(\|x_0\|)+\int_0^t\gamma(|u(\tau)|)\,d\tau$$

holds. Point-wise integral-to-integral conjecture. Then the system is ISS. This is proposed as an alternative route to proving ISS under point-wise dissipation; its resolution is not supplied.