On a multiplicative function on the set of shifted primes

It is proved that if $f(n)$ is a multiplicative function taking a value $\xi$ on the set of primes such that $\xi^3=1$, $\xi\ne1$ and $f^3(p^r)=1$ for $r\ge2$, then there exists $\theta\in(0,1)$, for which
$$ \biggl|\sum_{p\le x}f(p+1)\biggr|\le\theta\pi(x), $$
where
$$ \pi(x)=\sum_{p\le x}1. $$