GES under point-wise dissipation for time-delay systems

Let $f_0$ be the vector field of the autonomous time-delay system, Lipschitz on bounded sets, with $f_0(0)=0$. Let $V:\mathcal X^n\to\mathbb R_{\geq0}$ be a Lyapunov–Krasovskii functional (LKF), and let $\underline a,\overline a,a>0$ satisfy, for every $\phi\in\mathcal X^n$,

$$\underline a|\phi(0)|^2\leq V(\phi)\leq\overline a\|\phi\|^2,$$ $$D^+V(\phi,f_0(\phi))\leq-a|\phi(0)|^2.$$

GES under point-wise dissipation conjecture. Under these assumptions, the system is GES. The converse is known from the cited theorem, so the conjecture would characterize GES through point-wise dissipation without the additional growth-rate assumption currently required; its resolution is not supplied.