Erdős Problem 826

Reference: erdosproblems.com/826

theorem erdos_826 : (∃ C > 0, {n : ℕ | ∀ k ≥ 1, ↑((ArithmeticFunction.sigma 0) (n + k)) ≤ C * ↑k}.Infinite) ↔ sorry

Are there infinitely many $n$ such that, for all $k\geq 1$ $$ \tau(n + k) \ll k? $$