Arbitrarily slow decay in the logarithmically averaged Sarnak conjecture

In 2017 Tao proposed a variant Sarnak’s Möbius disjointness conjecture with logarithmic averaging: For any zero entropy dynamical system $(X,T)$, $\frac{1}{\log N}{\sum }_{n=1}^N\frac{f(T^{n}x)\mu (n)}{n}=o(1)$ for every $f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. Nonetheless, all of our examples satisfy the conjecture.