How to perform aggregation on nested of nested of nested field?

Oh wait, I found the typing mistake. This query worked:

GET order/_search
{
  "size": 0,
  "aggs": {
    "order": {
      "nested": {
        "path": "order_items.product"
      },
      "aggs": {
        "product_name" : {
          "terms": {
            "field": "order_items.product.name"
          },
          "aggs": {
            "avg_price": {
              "avg": {
                "field": "order_items.product.price"
              }
            }
          }
        }
      }
    }
  }
}