I'm facing some problems regarding the publication GREG using Swagger. I wonder if you have any way to make GREG recognize the JSON Schema, because I saw that it only recognizes XML Schema which ends up forcing me to use WADL.
I presume you are using G-Reg 4 series which does not have Swagger support. However, in G-Reg 5 series WSO2 have enabled this and you can find the latest version which is G-Reg 5.3.0 from here. Other than that, new G-Reg offers you some more very useful features to enhance SOA governance capabilities.
Adding a SOAP service using a Swagger
To enable json schema you have to create a handler. Please refer below sample media type handlers available in greg(carbon-registry)
Please find this useful article which will teach you how to create a simple handler.