Erdős Problem 288

Reference: erdosproblems.com/288

theorem Erdos288.erdos_288 : {I : Fin 2 → ℕ × ℕ | ∃ (n : ℕ), ∑ j : Fin 2, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑nⱼ)⁻¹ = ↑n}.Finite ↔ sorry

Is it true that there are only finitely many pairs of intervals $I_1$, $I_2$ such that $$ \sum_{n_1 \in I_1} \frac{1}{n_1} + \sum_{n_2 \in I_2} \frac{1}{n_2} \in \mathbb{N}? $$

theorem Erdos288.erdos_288.variants.i2_card_eq_1 : {(I, n₂) : (ℕ × ℕ) × ℕ | ∃ (n : ℕ), ∑ n₁ ∈ (Set.Icc I.1 I.2).toFinset, (↑n₁)⁻¹ + (↑n₂)⁻¹ = ↑n}.Finite ↔ sorry

This is still open even if $|I_2| = 1$.

theorem Erdos288.erdos_288.variants.k_intervals (k : ℕ) : {I : Fin k → ℕ × ℕ | ∃ (n : ℕ), ∑ j : Fin k, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑nⱼ)⁻¹ = ↑n}.Finite ↔ sorry

It is perhaps true with two intervals replaced by any $k$ intervals.

theorem Erdos288.erdos_288.variants.exists_k_gt_2 : ∃ k > 2, {I : Fin k → ℕ × ℕ | ∃ (n : ℕ), ∑ j : Fin k, ∑ nⱼ ∈ (Set.Icc (I j).1 (I j).2).toFinset, (↑nⱼ)⁻¹ = ↑n}.Finite ↔ sorry

Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition?