Erdős Problem 536

Reference: erdosproblems.com/536

theorem Erdos536.erdos_536 : (∀ ε > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, ε * ↑N ≤ ↑A.card → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, {a, b, c}.card = 3 ∧ a.lcm b = b.lcm c ∧ b.lcm c = a.lcm c) ↔ sorry

Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be distinct $a,b,c\in A$ such that $$[a, b]=[b, c]=[a, c],$$ where $[\cdot, \cdot]$ denotes the least common multiple?