That outcome is frustrating, and the question you're asking is a genuinely hard one to answer cleanly, for anyone.
Here is why the comparison data you're looking for is difficult to get: staged and unstaged listings almost never share identical conditions. Same street, same week, same price point, same days on market, same buyer pool, that combination basically never exists in a clean pair. Most of the published staging research compares averages across large samples, and at the $220k price point in a mid-market Midwest city, the sample gets thin fast. So you are probably not missing some obvious study. The clean data largely does not exist.
What I can say about your specific situation: $3,200 is roughly 1.4% of your original list price. If staging had held your price at $229k, it would have paid for itself about twice over. It did not, but we cannot know whether an empty house would have closed faster, slower, or at the same number. The market softening you described in weeks four through eight is a real variable that staging could not fix. Staging helps buyers visualize a finished life in a space; it does not manufacture demand when buyer activity drops.
The price point question you raised is worth sitting with. Some agents who work the $200k to $250k range in secondary markets will tell you buyers at that level are less influenced by furniture and more focused on condition and price. I would not treat that as settled fact, but it is a conversation worth having with a local agent who has data on your specific zip code.
One thing worth checking: what did your DOM (days on market, meaning the number of days between listing and accepted offer) look like against unstaged listings that closed in Dayton in that same window?
What did your listing agent say about why traffic dropped after week three?