The pointwise operator-norm divergence conjecture for stable LTI systems

Let $\Lambda = \{\lambda_n\}_{n\in\mathbb{Z}}$ be a complete interpolating sequence for $\mathcal{PW}^{2}_{\pi}$. Let $\mathrm{H}:\mathcal{PW}^{1}_{\pi}\to\mathcal{PW}^{1}_{\pi}$ be a stable LTI system, let $\mathrm{H}_N$ be its sampling-based approximation, and for $t\in\mathbb{R}$ and $\beta\in(0,1]$ define

$$\|\mathrm{H}_N\|_{t,\beta}=\sup\{ |(\mathrm{H}_Nf)(t)|: f\in\mathcal{PW}^{1}_{\beta\pi},\ \|f\|_{\mathcal{PW}^{1}_{\beta\pi}}\leq 1\}.$$

Pointwise operator-norm divergence conjecture. For every $t\in\mathbb{R}$, there exists a stable LTI system $\mathrm{H}:\mathcal{PW}^{1}_{\pi}\to\mathcal{PW}^{1}_{\pi}$ such that, for every $\beta\in(0,1]$,

$$\lim_{N\to\infty}\|\mathrm{H}_N\|_{t,\beta}=+\infty.$$

This would give a negative answer to the question of whether a subsequence of the approximation operators can converge globally uniformly to $\mathrm{H}f$ for every $f\in\mathcal{PW}^{1}_{\pi}$. The source notes that only a limsup divergence result is currently known.