A number $n$ is highly composite if $\tau(m)<\tau(n)$ for all $m<n$, where $\tau(m)$ counts the number of divisors of $m$. Let $Q(x)$ count the number of highly composite numbers in $[1,x]$.

Is it true that

$$Q(x)\gg_k (\log x)^k$$

for every $k\geq 1$?