Problem with term lookup filter

gistfile1.txt

curl -X POST 'localhost:9200/test_index/Task/test_task?routing=15793206719&refresh=true' -d '{
  "name": "a task",
  "index_ids": [
    123
  ]
}'

curl -X POST 'localhost:9200/test_index/Domain/test_domain?routing=15793206719&refresh=true' -d '{
  "public_index_ids": [
    123
  ]
}'


# Find all tasks that have an index_id of "123". This returns "test_task"
curl -X POST 'localhost:9200/test_index/Task/_search?routing=15793206719' -d '{
  "filter": {
    "terms": {
      "index_ids": [
        123
      ]
    }
  }
}'

# Find all tasks that have an index_id in test_domain.public_index_ids. This returns no hits.
curl -X POST 'localhost:9200/test_index/Task/_search?routing=15793206719' -d '{
  "filter": {
    "terms": {
      "_cache_key": "my_cache_key",
      "index_ids": {
        "index": "test_index",
        "type": "Domain",
        "id": "test_domain",
        "path": "public_index_ids"
      }
    }
  }
}'