Erdős Problem 700: Iii

Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$. (c)* Is it true that, for every composite $n$, $f(n) \ll_A n/(\log n)^A$ for every $A > 0$?

Erdős–Szekeres [ErSz78] prove the weaker bound $f(n) \le (1 + o(1)) n/\log n$ (the case $A = 1$). Here $f(n) \ll_A n/(\log n)^A$ is spelled out as: for every A > 0 there is a constant C (depending on A) with f(n) ≤ C · n/(log n)^A for every composite n.