Erdős Problem 383

Reference: erdosproblems.com/383

theorem erdos_383 : (∀ (k : ℕ), {p : ℕ | Nat.Prime p ∧ (∏ i ∈ Finset.Icc 0 k, (p ^ 2 + i)).maxPrimeFac = p}.Infinite) ↔ sorry

Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $$ \prod_{i = 0}^k (p ^ 2 + i) $$ is $p$?