Harper's conjecture on moments of Möbius character and zeta sums

Let $\mu(n)$ denote the Möbius function. For a large prime $r$, let $\chi$ range over Dirichlet characters modulo $r$; let $T$ be large, and let $q$ satisfy $0\leq q\leq 1$. For any fixed $A>0$, consider the averaged $2q$-moments of the Möbius character sums and of the corresponding Dirichlet polynomials on the unitary line.

Harper's conjecture. For all $0\leq q\leq 1$ and any fixed $A>0$, for large prime $r$,

$$\frac{1}{r-1}\sum_{\chi\;\text{mod}\;r}\left|\sum_{n\leq x}\mu(n)\chi(n)\right|^{2q}\ll_A\left(\frac{x}{1+(1-q)\sqrt{\log\log x}}\right)^q\qquad\forall\,x\leq r^A,$$

and, for large real $T$,

$$\frac{1}{2T}\int_{-T}^{T}\left|\sum_{n\leq x}\mu(n)n^{it}\right|^{2q}\,dt\ll_A\left(\frac{x}{1+(1-q)\sqrt{\log\log x}}\right)^q\qquad\forall\,x\leq T^A.$$

This conjectural extension would imply Möbius cancellation in intervals of length just $o(\sqrt{x})$ and in arithmetic progressions with modulus just larger than $\sqrt{x}$, including the Möbius analogue of Legendre's conjecture. It remains open; the paper notes that the corresponding Liouville-function character estimate is supported by work of Wang and Xu under the Generalised Riemann Hypothesis and a suitable Ratios Conjecture.