Is it true that for every completely multiplicative function $f:\mathbb N\to\mathbb C$ with $|f(n)|=1$ for all $n$, the logarithmic average $$\lim_{N\to\infty}\frac1{\log N}\sum_{n=1}^N\frac1nf(n)\quad\text{exists?}$$

In Elliott's book "Probabilistic number theory", a theorem attributed to Delange, Wirsing, Halasz (Theorem 6.3) describes exactly when a multiplicative function taking values in the unit disk has a (Cesaro) mean, and what is the value of the mean when it exists. It seems that the main obstacle to the existence of the (Cesaro) mean exist are the multiplicative functions $n^{it}$. On the other hand, it is easy to check that the logarithmic average of any such function exists (and is 0).

If the series $\sum_{p\text{ prime}}\frac{1-f(p)}p$ converges, then the answer is yes (this follows from the aforementioned theorem). On the other hand, an application of the Turan-Kubillius inequality shows that if for some $\epsilon>0$ the set $S:=\{p\text{ prime}:|f(p)-1|>\epsilon\}$ is divergent (in the sense that $\sum_{p\in S}\frac1p=\infty$), then again the answer to the question is yes (and the limit is $0$). However I don't see how to handle the general case.