Let $f(k)$ be the minimal value of $n_k$ such that there exist $n_1 < n_2 < \dots < n_k$ with $$1 = \frac{1}{n_1} + \cdots + \frac{1}{n_k}.$$ Is it true that $$f(k) = (1 + o(1)) \frac{e}{e - 1} k ?$$

1∀ (f : ℕ → ℕ) (S : Set ℕ),2  S = {k | ∃ n, StrictMono n ∧ 0 ∉ Set.range n ∧ 1 = ∑ i, 1 / ↑(n i)} →3    (∀ k ∈ S,4        IsLeast {x | ∃ n, ∃ (_ : StrictMono n) (_ : 0 ∉ Set.range n) (_ : 1 = ∑ i, 1 / ↑(n i)), n (Fin.last k) = x}5          (f k)) →6      ∃ o, ∃ (_ : o =o[Filter.atTop] 1), ∀ k ∈ S, (1 + o k) * Real.exp 1 / (Real.exp 1 - 1) * (↑k + 1) ≤ ↑(f k)