Sorting not working in Search index - MongoDB Atlas

We are currently using MongoDB Atlas as our database solution and have a Indexing and query optimization requirement. We have created a query that partially fulfills our needs, but the results are not meeting our expectations.

Our Requirements:

1. Search: We need to full text search in a collection.
2. Date Range: The query should allow us to specify a date range.
3. Sort by Time: Results should be sorted by timestamp in descending order.
4. Skip and Limit (Pagination): We require the ability to skip through results and limit the number of results per page.

Here's the current index and query we've been using for your reference:

Indexing created in Mongodb atlas

{
  "mappings": {
    "dynamic": false,
    "fields": {
      "level": {
        "type": "string"
      },
      "message": [
        {
          "dynamic": true,
          "type": "document"
        },
        {
          "type": "string"
        }
      ],
      "timestamp": {
        "type": "date"
      }
    }
  },
  "storedSource": {
    "include": [
      "timestamp",
      "level",
      "message"
    ]
  }
}

Query

   [
      {
        "$search": {
          "index": "srchIndxLogs***",
          "compound": {
            "must": [
              {
                "range": {
                  "path": "timestamp",
                  "gte": "2023-08-14T18:30:00.000Z",
                  "lte": "2023-09-15T18:29:59.000Z"
                }
              },
              {
                "phrase": {
                  "query": "TEST062",
                  "path": ["level", "message"]
                }
              }
            ]
          },
          "sort": {
            "timestamp": -1
          }
        }
      },
      {
        "$skip": 0
      },
      {
        "$limit": 100
      },
      {
        "$project": {
          "_id": 1,
          "timestamp": 1,
          "level": 1,
          "message": 1
        }
      }
    ]

Sample data

{"_id":{"$oid":"64f17d25a71e2f001cc19107"},"timestamp":{"$date":{"$numberLong":"1693547813848"}},"level":"info","message":"TEST062 device connected to the server","meta":null}

Sorting not working as expected in this query and search index, Please give the solution on this.