Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?

1True ↔2  ∀ (𝓒 : ℕ → ℕ),3    (Set.range 𝓒).Finite →4      ∃ S,5        (∀ (i j : ℕ), i ≠ j → Disjoint (S i) (S j)) ∧6          ∀ (i : ℕ), (∀ n ∈ S i, 2 ≤ n) ∧ ∑ n ∈ S i, 1 / ↑n = 1 ∧ (𝓒 '' ↑(S i)).Subsingleton