Erdős Problem 873

Reference: erdosproblems.com/873

theorem erdos_873 : (∀ (a : ℕ → ℕ), ∀ ε > 0, 0 < a 0 → StrictMono a → ∃ (k : ℕ), ∀ X > 0, ↑↑(F✝ a X k) < ↑(X ^ ε)) ↔ sorry

Let $A = \{a_1 < a_2 <⋯\} ⊆ ℕ$ and let $F(A,X,k)$ count the number of $i$ such that $[a_i,a_{i+1}, … ,a_{i+k−1}] < X$, where the left-hand side is the least common multiple. Is it true that, for every $ϵ > 0$, there exists some $k$ such that $F(A,X,k) < X^ϵ$?