The strong divergence conjecture for LTI system approximations

Let $\Lambda = \{\lambda_n\}_{n\in\mathbb{Z}}$ be a complete interpolating sequence for $\mathcal{PW}^{2}_{\pi}$. Let $\mathrm{H}_N$ be the sampling-based approximation operators of a stable LTI system $\mathrm{H}:\mathcal{PW}^{1}_{\pi}\to\mathcal{PW}^{1}_{\pi}$. For each $\beta\in(0,1]$, let $f_\beta\in\mathcal{PW}^{1}_{\beta\pi}$. Strong divergence conjecture. There exists a stable LTI system $\mathrm{H}:\mathcal{PW}^{1}_{\pi}\to\mathcal{PW}^{1}_{\pi}$ such that, for every $\beta\in(0,1]$, there is an $f_\beta\in\mathcal{PW}^{1}_{\beta\pi}$ satisfying

$$\lim_{N\to\infty}\|\mathrm{H}_Nf_\beta\|_{\infty}=\lim_{N\to\infty}\max_{t\in\mathbb{R}}|(\mathrm{H}_Nf_\beta)(t)|=+\infty.$$

This would give a negative answer to the question of whether the approximation sequence diverges strongly. The claim is presented as an expected extension of the preceding operator-norm divergence discussion.