Convergence conjecture for general measurement procedures

Let $\{t_k\}_{k\in\mathbb{Z}}\subset\mathbb{R}$ be an ordered complete interpolating sequence for $\mathcal{PW}_{\pi}^{2}$, let $\phi_k$ be as defined in the source, and let $0<\sigma<\pi$. Convergence conjecture. There exists a sequence of continuous linear functionals $\{c_k\}_{k\in\mathbb{Z}}$ on $\mathcal{PW}_{\pi}^{1}$ such that, for every stable LTI system $T:\mathcal{PW}_{\pi}^{1}\to\mathcal{PW}_{\pi}^{1}$ and every $f\in\mathcal{PW}_{\sigma}^{1}$,

$$\lim_{N\to\infty}\sup_{t\in\mathbb{R}}\left|(Tf)(t)-\sum_{k=-N}^{N}c_k(f)(T\phi_k)(t)\right|=0.$$

This claim says that suitable measurement functionals and oversampling yield uniformly convergent system approximation; the source states that it is proved in the cited section, so its database status is solved.