Plotting records (from A071383 and A071385) suggests that an upper bound of the form $$ \log r_2(n)\ll\log^\alpha n $$ is plausible, perhaps for $\alpha$ around $0.6$ or $0.7$. Polynomial bounds seem too large, while polylog bounds seem too small.
Edit: See the answer by GH from MO which shows that $$ \frac{2\log n}{\log\log n} $$ should be used in place of $\log^\alpha n.$