Erdős Problem #448 — Dyadic Divisor Blocks and Divisor Density

Define $\tau^+(n)$ to be the number of integers $k$ such that $n$ has a divisor in $[2^k,2^{k+1})$; equivalently, $\tau^+(n)$ is the cardinality of the image of the divisors of $n$ under $d\mapsto \lfloor\log_2 d\rfloor$. Let $\tau(n)$ be the number of divisors of $n$. Is it true that, for every real $\epsilon>0$, the set

$$\{n\in\mathbb N: \tau^+(n)<\epsilon\,\tau(n)\}$$

has natural density $1$?