Square-root cancellation conjecture for the oscillatory phase average

Let $A_N(\gamma)$ be the oscillatory phase average associated with the centred layer moment, let $X$ be the relevant scale, and let $R$ be the corresponding correlation quantity. Square-root cancellation conjecture.

$$A_N(\gamma)\ll X^{-1/2+\varepsilon}$$

Equivalently,

$$R^2\ll N^{-1+\varepsilon}.$$

The bound is expected from square-root cancellation in the sum over $m$; numerical data give $A_N(\gamma_1)\sim X^{-0.59}$, below the rigorous ceiling $X^{-1/4}$ and consistent with the conjectured exponent.