You can have your sneakers independently authenticated by a reputable service and provide them with proof. It's a hit and miss what they are going to do.
Twice have I been able to have them release my money by providing an authenticity certificate (by email), but one time, they didn't even give me the chance to provide anything, and sent my item back with an email that they "waived" the $20 shipping fee for the inconvenience. Huh

?? If they truly believed it's fake, no need to waive anything and I should be on the hook for the return shipping costs. But on the other hand, if my item is authentic and I could provide proof, the right thing for them to do is to admit their mistake and release my funds. I sent them the certificate of authenticity anyway afterwards and told them that if they are a company with integrity (like they claim to be) they should make this right. Posh is a fairly easy place to sell and thin out my collection, but sometimes, they are completely full of it.