Erdős Problem 396

Reference: erdosproblems.com/396

theorem Erdos396.erdos_396 : (∀ (k : ℕ), ∃ (n : ℕ), n.descFactorial (k + 1) ∣ n.centralBinom) ↔ sorry

Is it true that for every $k$ there exists $n$ such that $$\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$$