What is the best known bound for the Mertens function along arithmetic progressions? More specifically, what is the best bound known for

$$\sum_{n<x}\mu(kn)$$

as $k,x\to\infty$. This paper of Lynelle Ye gives a very complete solution, but only under the RH so it is not much use to me. This paper about the Mertens function along arithmetic progressions $an+b$ is also promising, but it works under the assumption $\gcd(a,b)=1$ the whole time, meaning that $b=0$ can not be applied.

Working out simple bounds using Perron's formula does not look very hard, but seeing as this seems like an incredibly natural result to already have been proved it feels like it would not be the best idea to put all this time into proving a weaker version of a result that already exists.