Is there a sequence $1 \le d_1 < d_2 < \dots$ with density 1 such that all products $\prod_{u \le i \le v} d_i$ are distinct?

1True ↔2  ∃ d,3    StrictMono d ∧4      1 ≤ d 0 ∧5        (Set.range d).HasDensity 1 ∧6          Set.InjOn7            (fun x =>8              match x with9              | (u, v) => ∏ i ∈ Finset.Icc u v, d i)10            {(u, v) | u ≤ v}