On a conjecture on sums of multiplicative functions

We consider the following conjecture, which was made in 1970 by B. V. Levin and A. S. Fainleib; if $f\in W$, $f(p)\leqslant g(p)$, $\sum_{p\leqslant x}g(p)\ln p\sim\tau_gx$, and (2) is fulfilled with $\tau_f\ne0$, then (1) holds. We prove that this conjecture holds if $\tau_f\cdot\tau_g >0$. In the case $\tau_f\cdot\tau_g\leqslant0$ we construct a counterexample to the conjecture. The asymptotic behavior of the sum of values of the function is found by an analytic method.