Bege's conjecture on sums of Apostol's Möbius functions in arithmetic progressions

For integers $k\geq 2$ and $q\geq 1$, define the error term $E_{k,q}(x)$ by

$$\sum_{\substack{n\leq x\\(n,q)=1}}\mu_k(n)=A_{k,q}x+E_{k,q}(x),$$

where

$$A_{k,q}=\frac{\varphi(q)}{q}\prod_{p\nmid q}\left(1-2p^{-k}+p^{-(k+1)}\right).$$

Here $\varphi(q)$ is Euler's totient function, $\theta(q)$ is the number of positive squarefree divisors of $q$, and $D$ and $A$ are absolute positive constants. Bege's conjecture. For every real number $x\geq 3$ and integers $k\geq 2$ and $q\geq 1$,

$$E_{k,q}(x)\ll \theta(q)x^{1/k}\exp\left(-D\frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right).$$

In particular, for $q=1$ this gives the corresponding estimate for $E_k(x)$. If the Riemann Hypothesis is true, the conjecture further predicts the improved estimate

$$E_{k,q}(x)\ll \theta(q)x^{2/(2k+1)}\exp\left(A\frac{\log x}{\log\log x}\right),$$

with the analogous $q=1$ estimate for $E_k(x)$. The conjecture concerns cancellation in sums of Apostol's Möbius functions subject to a coprimality condition; the source attributes it to A. Bege, but the supplied text gives no evidence that either estimate has been proved or disproved.