Need to get all key value pairs from a JSON containing a specific character '/'

I have a specific json content for which I need to get all keys which contains the character / in their values.

JSON

{ "dig":  "sha256:d2aae00e4bc6424d8a6ae7639d41cfff8c5aa56fc6f573e64552a62f35b6293e",
 "name": "example",
 "binding": { 
   "wf.example.input1": "/path/to/file1",
   "wf.example.input2": "hello",
   "wf.example.input3":
     ["/path/to/file3",
      "/path/to/file4"],
   "wf.example.input4": 44
  }
}

I know I can get all the keys containing file path or array of file paths using query jq 'paths(type == "string" and contains("/"))'. This would give me an output like:

[ "binding", "wf.example.input1" ]
[ "binding", "wf.example.input3", 0]
[ "binding", "wf.example.input3", 1 ]

Now that i have all the elements that contains some file paths as their values, is there a way to fetch both key and value for the same and then store them as another JSON? For example, in JSON mentioned for this question, I need to get the output as another JSON containing all the matched paths. My output JSON should look something like below.

 { "binding":
   { "wf.example.input1": "/path/to/file1",
     "wf.example.input3": [ "/path/to/file3", "/path/to/file4" ] 
   }
 }