chaining searches fails to find expected result

[braingram]

Description of the problem

The docs mention the ability to "chain" searches:
https://asdf.readthedocs.io/en/latest/asdf/features.html#chaining-searches
The shown example works (if type_ is used instead of type) but other chains are not producing the expected result.

Example of the problem

>>> import asdf
>>> af = asdf.AsdfFile({"a": 1, "b": {"c": [1, 2, 3]}})
>>> af.search("c")  # expected
root (AsdfObject)
└─b (dict)
  └─c (list): [1, 2, 3]
>>> af.search("b")  # expected
root (AsdfObject)
└─b (dict): {'c': [1, 2, 3]}
>>> af.search("b").search("c")  # fails to find "c"
No results found

System information

asdf version: main
python version: 3.10
operating system: mac osx