ISS under point-wise dissipation for time-delay systems

Let $f$ be the vector field of the time-delay system on the history space $\mathcal X^n$. Let $V:\mathcal X^n\to\mathbb R_{\geq0}$ be an LKF, with $\alpha\in\mathcal K_\infty$ and $\gamma\in\mathcal N$, such that for every $\phi\in\mathcal X^n$ and $v\in\mathbb R^m$,

$$D^+V(\phi,f(\phi,v))\leq-\alpha(|\phi(0)|)+\gamma(|v|).$$

ISS under point-wise dissipation conjecture. These assumptions imply that the system is ISS. The converse is known, so this would provide an alternative characterization of ISS and remove the growth restrictions on the dissipation rate; its resolution is not supplied.