MongoDB $lookup on nested document

I'm new to mongo and struggling mightily with the following. I have 2 collections structured as below. For the life of me, I can't figure out how to do a $lookup on the school collection. Reading other posts, I'm definitely using ObjectId for the reference as well as the foreign field.

Below is my structure:

Alumni:

{
    "_id": "john",
    "items": [
        {
            "name": "John",
            "items": [
                {
                    "school": ObjectId("56de35ab520fc05b2fa3d5e4"),
                    "grad": true
                },
                {
                    "school": ObjectId("56de35ab520fc05b2fa00000"),
                    "grad": false
                }
            ]
        },
        {
            "name": "Johnny"
            // notice no nested items, this doc should still be included in result
        },
        {
            "name": "Jon",
            "items": [
                {
                    "school": ObjectId("56de35ab520fc05b2fa11111"),
                    "grad": false
                }
            ]
        }
     ]
}

Schools

{
    _id: ObjectId("56de35ab520fc05b2fa3d5e4"),
    name: "Some University",
    street: "ABC Boulevard"
}

What I'm looking to get:

{
    "_id": "john",
    "items": [
        {
            "name": "John",
            "items": [
                {
                    "school": ObjectId("56de35ab520fc05b2fa3d5e4"),
                    "grad": true,
                    "schoolInfo":     {
                        _id: ObjectId("56de35ab520fc05b2fa3d5e4"),
                        name: "Some University",
                        street: "ABC Boulevard"
                    }
                },
                {
                    "school": ObjectId("56de35ab520fc05b2fa00000"),
                    "grad": true,
                    "schoolInfo":     {
                        _id: ObjectId("56de35ab520fc05b2fa00000"),
                        name: "Another University",
                        street: "123 Boulevard"
                    }
                }
            ]
        },
        {
            name: "Johnny"
        },
        {
            "name": "Jon",
            "items": [
                {
                    "school": ObjectId("56de35ab520fc05b2fa11111"),
                    "grad": true,
                    "schoolInfo":     {
                        _id: ObjectId("56de35ab520fc05b2fa11111"),
                        name: "Some University",
                        street: "ABC Boulevard"
                    }
                }
            ]
         }
     ]
}

The query I've tried to no avail:

db.alumni.aggregate([
      {$match: {_id: 'john'}}, 
      {$lookup: {
                from: 'schools', 
                localField: 'items.items.school', 
                foreignField: '_id', 
                as: 'schoolInfo'}}
 ])

Any help would be greatly appreciated!