The strong divergence conjecture for arbitrary complete interpolating sequences

Let $\Lambda = \{\lambda_n\}_{n\in\mathbb{Z}}$ be an arbitrary complete interpolating sequence with generator $\varphi$, and let $\{\varphi_n\}_{n\in\mathbb{Z}}$ be the corresponding interpolation kernels. For $f\in\mathcal{PW}^{1}_{\pi}$, define

$$(\mathrm{A}_Nf)(t)=\sum_{n=-N}^{N}f(\lambda_n)\varphi_n(t).$$

Strong divergence conjecture. There exists an $f\in\mathcal{PW}^{1}_{\pi}$ such that

$$\lim_{N\to\infty}\|\mathrm{A}_Nf\|_{\infty}=\lim_{N\to\infty}\max_{t\in\mathbb{R}}\left|\sum_{n=-N}^{N}f(\lambda_n)\varphi_n(t)\right|=\infty.$$

Strong divergence is known for sampling patterns arising from sine-type functions of the specified form; the conjecture extends this phenomenon to arbitrary complete interpolating sequences.