The square-root cancellation conjecture for Fourier coefficients of cusp forms

Let $f$ be a cusp form whose Fourier coefficients are given by a multiplicative function $c:\mathbb{N}\to\mathbb{R}$. For each prime $p_n$, write $c(p_n)=|c(p_n)|\cos\theta_{p_n}$ with $\cos\theta_{p_n}=\pm1$, and define

$$C_N=\sum_{n=1}^N\cos\theta_{p_n}.$$

Square-root cancellation conjecture. As $N\to\infty$,

$$C_N=O(\sqrt{N}).$$

This is the cusp-form analogue of the preceding conjecture and is motivated in the paper by treating the signs of the prime Fourier coefficients as independent random choices. The source offers heuristic support rather than a proof.