jq: Getting two levels of keys

Question

I have some json data that looks like:

{
  "p": {
    "d": {
      "a" : {
        "r": "foo",
        "g": 1
      },
      "b": {
        "r": "bar",
        "g": 2
      }
    },
    "c": {
      "e": {
        "r": "baz",
        "g": 1
      }
    },
    ...
  }
}

I want something like:

{
  "d": [ 
    "a",
    "b"
  ],
  "c": [
    "e"
  ]
}

I can get the list of keys on the first level under "p" with jq '.p|keys', and the structure and keys on the second level with jq '.p|map(.|keys)', but I can't figure out how to combine it.