Erdős Problem 386

Reference: erdosproblems.com/386

theorem Erdos386.erdos_386 : sorry ↔ ∀ k ≥ 2, ∃ᶠ (n : ℕ) in Filter.atTop, k ≤ n - 2 ∧ ∃ (p : ℕ) (q : ℕ), n.choose k = ∏ i ∈ Finset.Ico p q, Nat.nth Nat.Prime i

Let $2 \le k \le n - 2$. Can $\binom{n}{k}$ be the product of consecutive primes infinitely often?