Erdős problem 448

Let $\tau(n)$ count the divisors of $n$ and $\tau^+(n)$ count the number of $k$ such that $n$ has a divisor in $[2^k, 2^{k+1})$. Is it true that, for all $\epsilon > 0$, $$\tau^+(n) < \epsilon \cdot \tau(n)$$ for almost all $n$?