Harper's beyond square-root cancellation conjecture for character twists

Let $c\in\{\mu,\lambda\}$, where $\mu$ is the Möbius function and $\lambda$ is the Liouville function. Let $r$ be a prime modulus, let $\mathbb E_\chi$ denote the average over Dirichlet characters modulo $r$, and let $T>0$. Fix a positive constant $A$. Harper's conjecture. As $x\to+\infty$, provided $1\leq x\leq r^A$, one has

$$\mathbb E_\chi\left|\sum_{1\leq n\leq x}c(n)\chi(n)\right|=o(\sqrt{x}),$$

and, if $1\leq x\leq T^A$ and $x\to+\infty$, then

$$\frac1T\int_0^T\left|\sum_{1\leq n\leq x}c(n)n^{it}\right|\,dt=o(\sqrt{x}).$$

Moreover, in both estimates the quantitative upper bound $O\!\left(\frac{\sqrt{x}}{(\log\log x)^{1/4}}\right)$ holds. This conjecture extends Harper's beyond-square-root cancellation phenomenon beyond the conductor-limited ranges $x\leq r$ and $x\leq T$; the paper studies consequences conditional on a suitable Ratios Conjecture.