Erdős Problem 289

Reference: erdosproblems.com/289

theorem erdos_289 : (∀ᶠ (k : ℕ) in Filter.atTop, ∃ (I : Fin k → Finset ℕ), (∀ (i : Fin k), 2 ≤ (I i).card ∧ ∃ (a : ℕ) (b : ℕ), 0 < a ∧ I i = Finset.Icc a b) ∧ ∑ i : Fin k, ∑ n ∈ I i, (↑n)⁻¹ = 1) ↔ sorry

Is it true that, for all sufficiently large $k$, there exists finite intervals $I_1, \dotsc, I_k \subset \mathbb{N}$ with $|I_i| \geq 2$ for $1 \leq i \leq k$ such that $$ 1 = \sum_{i=1}^k \sum_{n \in I_i} \frac{1}{n}. $$