Let $1\leq u_1 < u_2 < \cdots$ be the sequence of integers with at most $2$ prime factors. Is it true that $$\limsup_{k \to \infty} \frac{u_{k+1}-u_k}{\log k}=\infty?$$

1True ↔2  Filter.limsup3      (fun k =>4        (↑↑(Nat.nth (fun n => 0 < n ∧ ArithmeticFunction.cardFactors n ≤ 2) (k + 1)) -5            ↑↑(Nat.nth (fun n => 0 < n ∧ ArithmeticFunction.cardFactors n ≤ 2) k)) /6          ↑(Real.log (↑k + 1)))7      Filter.atTop =8    ⊤