How to project a specific index inside a multilevel nested array in mongodb

I have a particular field in my document which has a multilevel nested array structure. The document looks like something this

{
    "_id" : ObjectId("62171b4207476091a17f595f"),
    "data" : [ 
        {
            "id" : "1",
            "content" : [ 
                {
                    "id" : "1.1",
                    "content" : []
                }, 
                {
                    "id" : "1.2",
                    "content" : [ 
                        {
                            "id" : "1.2.1",
                            "content" : [ 
                                {
                                    "id" : "1.2.1.1",
                                    "content" : []
                                }
                            ]
                        }, 
                        {
                            "id" : "1.2.2",
                            "content" : []
                        }
                    ]
                }
            ]
        }
    ]
}

(The ids in my actual data is a random string, I have added a more defined id here just for readability)

In my application code the nesting level is controlled so it won't go more than 5 levels deep.

I need to project a particular object inside one of the deeply nested arrays.

I have all the information needed to traverse the nested structure. For example if I need to fetch the object with id "1.2.2" my input will look something like this:

[{id: 1, index: 0}, {id: 1.2, index: 1}, {id: 1.2.2, index: 1}]

In the above array, each element represents one level of nesting. I have the Id and the index. So in the above example, I know I first need to travel to index 0 at the top level, then inside that to index 1 , then inside that to index 1 again to find my object.

Is there a way I can only get the inner object that I want directly using a query. Or will I need to get the whole "data" field and do the traversal in my application code. I have been unable to figure out any way to construct a query that would satisfy my need.