Uniform boundedness conjecture for the Fourier interpolation basis functions

Let $f_n(x)$ denote the basis functions of the Fourier interpolation considered in the paper, for $n\in\mathbb{Z}_{\ge0}$ and $x\ge0$. Uniform boundedness conjecture. There exists $C>0$ such that

$$|f_n(x)|<C$$

for all $n\in\mathbb{Z}_{\ge0}$ and $x\ge0$. This is motivated by numerical computations of the first 10000 basis functions, which found very few values exceeding $1$ in absolute value and none exceeding $2$; no rigorous error bounds are currently provided for the numerical method.