Atlas search - Exclude part of the query from scoring and sorting

Question

I have a query that also checks whether a document has "valid" in a field, or this field does not exist. In both cases, I don't want this query to affect the scoring, so that if I sort by relevance (the default), documents without this field will also appear at the top. The documents looks like:

{
   "names": [...],
   "stats": [
       {
          "name": "...",
          "status": "valid"
       }
   ]
}

There are some documents with names but without stats, and this is as good as a valid status, however the documents with the stats appear first.

My query:

db.collection.aggregate([
  {
    $search: 
    {
      index: [index name],
      "compound": {
        "should": [
          [some should conditions that should affect the score]
        ],
        "minimumShouldMatch": 1,
        must: [
          {
            compound: {
              should: [
                {
                  phrase: {
                    path: 'stats.status',
                    query: "valid"
                  }
                },
                {
                  compound: {
                    mustNot: [{
                      exists: {
                        path: 'stats'
                      }
                    }]
                  }
                }
              ],
              "minimumShouldMatch": 1,
            },
          },
        ]
      },
    }
  },
  {
    $addFields: {
        score: {$meta: 'searchScore'}
    }
  },
])