Logarithmic lower-bound conjecture for complete interpolating sequences

Let $\{t_k\}_{k\in\mathbb{Z}}\subset\mathbb{R}$ be an ordered complete interpolating sequence for $\mathcal{PW}_{\pi}^{2}$. Logarithmic lower-bound conjecture. There exists a positive constant $C_{12}$ such that

$$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\sum_{k=-N}^{N}e^{\mathrm{i}\omega t_k}\widehat{\phi_k}(\omega_1)\right|\,d\omega_1\geq C_{12}\log(N)$$

for all $\omega\in[-\pi,\pi]$ and all $N\in\mathbb{N}$. The conjecture is presented as a possible explanation for pointwise-sampling divergence, while the supplied text gives no resolution evidence; it therefore remains open in the database.