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erdos_288.variants.exists_k_gt_2

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

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Lean 4 Statement 
theorem erdos_288.variants.exists_k_gt_2 : answer(sorry) ↔
    ∃ k > 2, Set.Finite { I : Fin k → ℕ+ × ℕ+ |
      ∀ j, (I j).1 ≤ (I j).2 ∧
        ∃ n : ℕ+, (∑ j : Fin k, ∑ nⱼ ∈ Set.Icc (I j).1 (I j).2, (nⱼ⁻¹ : ℚ)) = n }
ID: ErdosProblems__288__erdos_288.variants.exists_k_gt_2
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