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 $n$?

Always $\tau^+(n) \le \tau(n)$: the occupied dyadic blocks are the image of the divisor set under Nat.log 2, and an image has at most as many elements as its source. This is what makes the $\epsilon < 1$ comparison in the problem meaningful.