Smooth-support mean-value conjecture for completely multiplicative functions

Let $x$ be sufficiently large. Consider a real-valued completely multiplicative function $f$ satisfying $|f(n)|\leq 1$ for every positive integer $n$, and suppose that $f(p)=0$ for every prime $p>x^\varepsilon$, where $\varepsilon>0$ is an absolute constant. Smooth-support mean-value conjecture. There exists $\varepsilon>0$ such that, for every such function $f$,

$$\sum_{n\leq x}f(n)\ll \frac{x}{\log\log x}\exp\left(\sum_{p\leq x}\frac{f(p)}{p}\right).$$

This estimate would improve the theorem preceding the conjecture on the probability that $\sum_{n\leq x}f(n)/n<0$. It concerns completely multiplicative functions supported on integers whose prime factors are at most $x^\varepsilon$, and its asserted uniformity is the key unresolved input.