I work in logistics and i have to say you may have a bad case here because you signed off on the delivery.
That declared you received the goods in good condition so the burden of proof is on your side to show it happened during transport. The picture of the unopened box with the hole might help in this case.
Since you claimed this case immediately you at least didn't miss any deadline, that's good.
In my experience companies usually don't make a big deal and replace items because they know that these things happen (you wouldn't believe how rough sometimes goods are handled in warehouses, even delicate things like electronics) and they want to avoid a bad rep (replacing it costs them a fraction of what you paid for it and they will most likely claim this to their distributor/forwarder and hand it over to the insurance company).
Good luck.. the fact that the lady didn't immediately block your request is a good sign but don't get your hopes up yet.