From time to time it will just refuse to pair to any headphones, even the ones that it was paired before.
The solution was to do a firmware update for Avantree Oasis Plus following the official tutorial. Please note they want to inform support before the update so you don’t loose the warranty.
The tutorial is here https://avantree.com/product-support/BTTC-500P-GRY#general-how-to-conduct-firmware-update-dfu
In short, you download the new firmware, install some software, put your Avantree Oasis Plus is „firmare update mode” and then send the new firmware to the device.