MongoDB $push to nested child - recursive data structure

I have the following schema and route set up. I cannot post request further into the nested child. Ideally, I want to be able to post request like the following (post to 3 results in id 4 child). Any help is appreciated.

Edit: Is it even possible without recursively findOne through each index like findOne at top -> findOne at child -> findOne at child2 till it finds matching id? Is restructuring database to make it flat and adding parentID key the only way?

[
  {
    {
      id:1,
      child: [
        {
          id:3,
            child: [
              {
                id:4,
                child:[]
              }
           ]
         }
       ]
    }
    id:2,
    child:[]
    }
]

MongoDB JSON

[
    {
        "_id": "6050036c5138b72ad07267e5",
        "title": "dddd",
        "completed": true,
        "__v": 1,
        "child": [
            {
                "_id": "60518f6c497c2f59a4161ac5",
                "title": "gooooood",
                "completed": false,
                "child": [] 
//Cannot post request here with _id="60518f6c497c2f59a4161ac5"
            }
        ]
    }
]

task-model.ts

import mongoose from "mongoose";

export const taskSchema = new mongoose.Schema(
  {
    title: { type: String, required: true },
    completed: { type: Boolean, required: true },
  },
);

taskSchema.add({ child: [taskSchema] });

export const Task = mongoose.model("Task", taskSchema);

route.ts

router.route("/:id").post((req, res) => {
  Task.updateOne(
    { _id: req.params.id },
    { $push: { child: req.body.child }},
  )
    .then(() => res.json("task added"))
    .catch((err) => res.status(400).json("Error" + err));
});