Erdős Problem #698 — Growing common divisors of binomial coefficients

Does there exist a function $h:\mathbb N\to\mathbb N$ with $h(n)\to\infty$ as $n\to\infty$ such that, for all natural numbers $n,i,j$ satisfying $2\le i<j$ and $j\le n/2$, one has

$$h(n)\le \gcd\!\left(\binom ni,\binom nj\right)?$$