When the capitalization of a file name deployed via the plugin changes, it is not accessible under the new capitalization, but only the old. Also the capitalization in the branch is not changed and stays wrong. Most likely this has to do with GitHub using a case insensitive file system.
1st run: a file called AirPlayDevice.html is pushed to gh-pages branch and can be accessed under AirPlayDevice.html. Great!
2nd run: the file name's capitalization is changed to AirplayDevice.html. When trying to access the file under AirplayDevice.html, a 404 is returned. When trying using the (now wrong!) name AirPlayDevice.html the correct file is returned. In the gh-branches branch, the capitalization is now wrong (no lowercase p).
Note that the file is generated freshly before calling github-pages-deploy-action. So the problem definitely occurs when merging changes into the target branch.