Conjectured bounds for real-variable Kloosterman sums in dimensions 3 and 4

Let $B_{d,n}(r)$ and $\widetilde{B}_{d,n}(r)$ be the functions appearing in the radial Fourier interpolation formulas for dimensions $d=3,4$, with $n$ an appropriate positive integer, $r\in\mathbb R$, and constants $c>0$ and $\varepsilon>0$. Conjectured bounds. If $r^2\geq c$, then

$$|B_{4,n}(r)|,\ |\widetilde{B}_{4,n}(r)|\ll_{\varepsilon,c} n^{1/2+\varepsilon},$$

while

$$|B_{3,n}(r)|,\ |\widetilde{B}_{3,n}(r)|\ll_{\varepsilon,c} n^{1/4+\varepsilon}.$$

These bounds would strengthen the proved estimates $n^{3/4+\varepsilon}$ in dimension $4$ and $n^{1/2+\varepsilon}$ in dimension $3$, giving sharper control of the coefficients in the radial Fourier interpolation formulas. They are presented as an optimistic conjecture and remain open.