Is there some absolute constant $C > 0$ such that $$\sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} \leq C$$ for all $n$?

Erdos, Graham, Ruzsa, and Straus proved that if $$f(n) = \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p}$$ then there is some constant $c < 1$ such that for all large $n$ $$f(n) \leq c \log\log n.$$

[EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., _On the prime factors of $\binom{2n}{n}$_. Math. Comp. (1975), 83-92.