Logarithmic square-norm conjecture for the Fourier interpolation basis functions

Let $f_n(x)$ be the Fourier interpolation basis functions. Logarithmic square-norm conjecture. For $n>1$, one has

$$\int_0^\infty |f_n(x)|^2\,dx\asymp\log n.$$

The lower bound is already known from the result cited in the paper, so the conjecture concerns the corresponding upper bound. If true, it would make the lower bound in the paper's square-norm theorem optimal up to a multiplicative constant.