No relaxation gap for peak estimation of time-delay systems

Let $T$ and $\tau$ satisfy $T > \tau > 0$, and let $f(t,x_0,X)$ denote the image of $X$ under $x_1 \mapsto f(t,x_0,x_1)$ for fixed $(t,x_0)$. Assume that A1–A5 hold and that

$$f(t,x_0,X)\text{ is convex for all fixed }t\in[0,T],\ x_0\in X.$$

No-relaxation-gap conjecture. There is no relaxation gap between the trajectory formulation

$$and the measure formulation$$

, so that $p^*=P^*$.

This conjecture concerns whether the occupation-measure relaxation exactly captures the peak-estimation problem for time-delay systems under the stated convexity condition. The source motivates it by results on relaxation for differential equations, but provides no resolution.