Moment conjecture for k-free numbers in short arithmetic progressions

Let $X>1$, let $q$ be a positive integer, and let $a$ range over residue classes relatively prime to $q$. Define the error term $E(X,q,a)$ by

$$\sum_{\substack{n\leq X\\ n\equiv a\!\!\!\!\!\pmod q}}\mu_k(n)=\frac{A_q}{\varphi(q)}X+E(X,q,a),$$

where

$$A_q:=\frac{\varphi(q)}{q}\prod_{p\,\nmid\,q}\left(1-\frac{1}{p^k}\right).$$

For $ell\geq 1$, define the $ell$-th moment by

$$\mathcal{M}(X,q,\ell)=\sideset{}{^{\ast}}\sum_{a\!\!\!\pmod q}\!\!\!E(X,q,a)^\ell,$$

where the sum is over the reduced residue classes modulo $q$. Moment conjecture. For integers $k\geq 2$ and $ell\geq 1$, there exist $\delta=\delta_{k,\ell}>0$, $\eta=\eta_{k,\ell}>0$, and a positive multiplicative function $c=c_{k,\ell}$ such that

$$\left|\mathcal{M}(X,q,\ell)-c(q)\varphi(q)\left(\frac{X}{q}\right)^{\frac{\ell}{2k}}\right|\leq \varphi(q)\left(\frac{X}{q}\right)^{\frac{\ell}{2k}-\eta},$$

for every $q$ and $X$ satisfying

$$X^{1-\delta}\leq q\leq X^{1-\varepsilon},\qquad X\geq 2.$$

Moreover, if $ell$ is even, then $c(q)$ is bounded above and below by positive constants depending only on $k$ and $ell$, while if $ell$ is odd, then $c\equiv 0$. The conjecture predicts the moments suggested by the probabilistic model for the distribution of $k$-free numbers among reduced residue classes, in the range where $q$ is close to $X$; the source gives no resolution, so its status remains open.