Is there some $h(n)\to \infty$ such that for all $2\leq i<j\leq n/2$ $$\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?$$

Is there some $h(n)\to \infty$ such that for all $2\leq i<j\leq n/2$ $$\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?$$

This was resolved by Bergman [Be11], who proved that for any $2\leq i<j\leq n/2$ $$\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \gg n^{1/2}\frac{2^i}{i^{3/2}},$$ where the implied constant is absolute.