Goldmakher's conjecture on logarithmic sums of multiplicative functions

Let $\mathcal{F}$ denote the collection of all $1$-bounded, completely multiplicative functions. For $f,g\in\mathcal{F}$ and $z\geq 1$, define the pretentious distance by

$$\mathbb{D}(f,g;z):=\left(\sum_{p\leq z}\frac{1-\operatorname{Re}(f(p)\overline{g(p)})}{p}\right)^{1/2}.$$

Goldmakher's conjecture. Let $f\in\mathcal{F}$. Then for any $1\leq y\leq x$,

$$\sum_{\substack{n\leq x\\ p\mid n\Rightarrow p\leq y}}\frac{f(n)}{n}\ll 1+(\log y)e^{-\mathbb{D}(f,1;y)^2}.$$

The conjecture relates the size of logarithmic partial sums to the pretentious distance from the constant function $1$, asserting that large or unbounded sums can occur only when $f$ pretends to be $1$. The paper states that the conjecture is disproved by constructing a function for which the corresponding ratio is unbounded.