Plotting records (from <a href="https://oeis.org/A071383">A071383</a> and <a href="https://oeis.org/A071385">A071385</a>) 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 <a href="https://mathoverflow.net/a/191354/6043">the answer by GH from MO</a> which shows that
$$
\leq\frac{(\log 2+o(1))\log n}{\log\log n}
$$
should be used in place of $\ll\log^\alpha n.$