The constant-function mimicry conjecture for weighted multiplicative sums

Let $\mathcal{F}$ be the class of multiplicative functions under consideration, let $\mathcal{S}(y)$ denote the integers whose prime factors are at most $y$, and let $\mathbb{D}(f,1;y)$ be the pretentious distance from $f$ to the constant function $1$ up to $y$. For $f\in\mathcal{F}$ and $2\leq y\leq x$, consider the $y$-smooth weighted sum

$$\sum_{\substack{n\leq x\\ n\in\mathcal{S}(y)}}\frac{f(n)}{n}.$$

Constant-function mimicry conjecture. One has

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

This conjecture asserts that the only genuine obstruction to a bounded weighted sum is mimicry of the constant function $1$, rather than mimicry of functions $n^{it}$.