On one hand, as the nice sketch provided below by tsaf confirms, the final proof is quite simple and elementary - indeed, if one was given this problem in a maths competition (so therefore expected a short simple solution existed) I'd guess that something like the below would be produced. On the other hand, if something like this worked, then surely the combined talents of Burr, Erdős, Graham, and Li would have spotted it.
Normally, this would make me suspicious of this short proof, in that there is overlooked subtlety. But (a) I can't see any and (b) the proof has been formalised in Lean, so clearly it just works!
Perhaps this shows what the real issue in the [BEGL96] conjecture is - namely the removal of 1 and the addition of the necessary gcd condition. (And perhaps at least some subset of the authors were aware of this argument for the easier version allowing 1, but this was overlooked later by Erdős in [Er97] and [Er97e], although if they were aware then one would hope they'd have included this in the paper as a remark.)
At the moment I'm minded to keep this as open, and add the gcd condition in the main statement, and note in the remarks that the easier (?) version allowing 1 and omitting the gcd condition, which was also asked independently by Erdős, has been solved."
The commentator is saying: "I can't believe this famous problem was solved so easily. I would have thought it was a fake proof, but the computer verified it. It turns out the solution works because it addresses a slightly different set of constraints (regarding the number 1) than what Erdős originally struggled with. (Generated by Gemini)