Interior-region decay conjecture for the Fourier interpolation basis functions

Let $f_n(x)$ be the Fourier interpolation basis functions, and let $\varepsilon,\delta>0$. Consider the region $\varepsilon n\le x\le n-\sqrt n$. Interior-region decay conjecture. For any $\varepsilon,\delta>0$, one has

$$f_n(x)=O_{\varepsilon,\delta}\!\left(n^{-1/2+\delta}\right)$$

for $\varepsilon n\le x\le n-\sqrt n$. The paper notes that this improves the preceding estimate $f_n(x)=O(n^{-0.392})$ in the stated regime and would follow from the exponent pair conjecture.