[Kikibosum]

In the end, she tells me that she can't do anything, nor her colleagues, and that she couldn't launch an investigation with UPS if the signature was signed

My goodness, that's total BS.

Companies open external Investigations with the carrier to cross-check everything relative to the claim, signatures Included. Tell her precisely what I've said and to open an Investigation asap.

At this point she told me that all she could advise me to do was to contact UPS directly, show them that the signature was not mine

Again, they're responsible for an Investigation.

and at that point, after a conclusive investigation on their part, UPS would contact New Balance and New Balance would be able to refund me. I find this absurd, since in the end I have to initiate the investigation personally, but this signature really doesn't belong to me. Do I do it? Do you think UPS will accept the claim?

See what happens with what I've stated above.

If NB still fail to use common sense, that's when you contact UPS. Be sure to keep this Information on hand:

UPS would contact New Balance and New Balance would be able to refund me

That's your refund reassurance.

No matter how many times I call back and tell them everything you tell me, it's always the same thing, they can't do anything. I even told them that given the reputation on some online review sites with this recurring problem, I will contribute to these bad reviews but they don't care. I opened a claim with UPS by checking the box: "Marked delivered but never received" and not "The signature does not belong to me". I'm sticking to the so-called NB version.