If $a/b \in \mathbb{Q}_{>0}$ and $b$ is squarefree, can $a/b$ always be written as a finite sum of reciprocals of distinct products of two distinct primes?

1∀ (m : ℕ),2  0 < m →3    ∃ k > 0,4      ∃ n,5        n 0 = 1 ∧6          ∀ (i : ℕ) (hik : i < k),7            n ⟨i, ⋯⟩ < n ⟨i + 1, ⋯⟩ ∧8              (∀ i ∈ Finset.Icc 1 (Fin.last k),9                  ArithmeticFunction.cardDistinctFactors (n i) = 3 ∧ ArithmeticFunction.cardFactors (n i) = 3) ∧10                ↑m = ∑ i ∈ Finset.Icc 1 (Fin.last k), 1 / ↑(n i)