Erdős Problem 248

Reference: erdosproblems.com/248

theorem Erdos248.erdos_248 : (∃ C > 0, {n : ℕ | ∀ k ≥ 1, ↑(ArithmeticFunction.cardDistinctFactors (n + k)) ≤ C * ↑k}.Infinite) ↔ sorry

Are there infinitely many $n$ such that $\omega(n + k) \ll k$ for all $k \geq 1$? Here $\omega(n)$ is the number of distinct prime divisors of $n$.