Erdős Problem 938

Reference: erdosproblems.com/938

theorem erdos_938 : {P : Finset ℕ | (↑P).IsAPOfLength 3 ∧ ∃ (k : ℕ), P = {Nat.nth Nat.Powerful k, Nat.nth Nat.Powerful (k + 1), Nat.nth Nat.Powerful (k + 2)}}.Finite ↔ sorry

Let $A=\{n_1 < n_2 < \cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$). Are there only finitely many three-term progressions of consecutive terms $n_k,n_{k+1},n_{k+2}$?