[mothered]

I remember them telling me and I quote: An investigation will be made and we will refund your order.
Just be patient.

Then they refunded me BEFORE they started the investigation.

They provided an advanced refund pending the results of their Investigation.

The proof of delivery is a pdf from UPS with my delivery address written on it along with the package info: weight, name, sender...

There is also written at the end received by NAME, FIRST NAME and phone number that's all.

So what? By no means Is any of that proof of delivery.

They're simply the details of the consignment, and not evidence confirming receipt of goods.

In short, no concrete proof of delivery.

Not only that but as said, nothing confirming that you personally received your package.

As such, you have sufficient grounds to claim nonreceipt of goods, Inclusive of support to hit a chargeback.

[Defeat]

The problem is that I don't know if I should continue to ignore them or if not, I thought of sending an email playing the legitimate stunned customer and asking them to stop harassing, threatening and falsely accusing me.

We're going to need more details. If it was a simple warning that's fine but if they said they're investigating with an intention to pursue legal matters, that's something else.