The Method used in your Postman request is of type 'POST', whereas Persisted queries can work only with GET method.
Hence all parameters have to be a part of URL, separated by ';' or %3b.
Example: append the %3bpaths=%7B"_expressions":[%7B"value":"/content/site1"%7D,%7B"value":"/content/site2"%7D,%7B"value":"/content/site3"%7D] in the URL