It is known that $$ \exp\left\{k(1/1.71\cdots+o(1))\right\} < H(k) < \exp\left\{k(1/1.13862\cdots+o(1))\right\}, $$ where $H(k)$ is the $k^{th}$ highly composite number.
Question: Does the number of divisors of the highly composite numbers $d(H(k))$ achieve the growth rate given by $$ \lim_{n \rightarrow \infty} \sup \frac{\log d(n)}{\frac{ \log n}{\log \log n}}=\log 2? $$ To be more precise let $n_k=H(k).$ Does $$ \lim_{k \rightarrow \infty} \sup \frac{\log d(n_k)}{\frac{ \log n_k}{\log \log n_k}}=\log 2? $$