In a related and highly relevant comment thread, Mike Miller pointed me to this [preprint][1] of Lipyanskiy.  I'm sure there are arguments which work, such as what Joshua and Dmitri and I discuss in the comments.  But before I forget I'd like to point to Lipyanskiy's work, especially section 5, and mark the question as answered.


  [1]: https://arxiv.org/abs/1409.1121