Oh, well I figured it out 2 minutes after posting this haha.
Maybe this will help someone else. The SpecificDataJsonSchema property works fine, but the value has to look like this:
{ \"$schema\": \"http://json-schema.org/draft-07/schema\", \"$id\": \"http://example.com/example.json\", \"type\": \"object\", \"title\": \"0330 Test Schema\", \"description\": \"0330 Test Schema\", \"default\": {}, \"examples\": [ { \"Name\": \"Example string\", \"Address\": \"Example string\" } ], \"required\": [], \"properties\": { \"Name\": { \"$id\": \"#/properties/Name\", \"type\": \"string\", \"title\": \"The Name schema\", \"description\": \"Name\", \"examples\": [ \"Example string\" ] }, \"Address\": { \"$id\": \"#/properties/Address\", \"type\": \"string\", \"title\": \"The Address schema\", \"description\": \"Address\", \"examples\": [ \"Example string\" ] } }, \"additionalProperties\": false}
The full body:
{
"Name": "AAAAAAA",
"SpecificDataJsonSchema": "{ \"$schema\": \"http://json-schema.org/draft-07/schema\", \"$id\": \"http://example.com/example.json\", \"type\": \"object\", \"title\": \"0330 Test Schema\", \"description\": \"0330 Test Schema\", \"default\": {}, \"examples\": [ { \"Name\": \"Example string\", \"Address\": \"Example string\" } ], \"required\": [], \"properties\": { \"Name\": { \"$id\": \"#/properties/Name\", \"type\": \"string\", \"title\": \"The Name schema\", \"description\": \"Name\", \"examples\": [ \"Example string\" ] }, \"Address\": { \"$id\": \"#/properties/Address\", \"type\": \"string\", \"title\": \"The Address schema\", \"description\": \"Address\", \"examples\": [ \"Example string\" ] } }, \"additionalProperties\": false}",
"Description":"test 2",
"Id":951
}