I have just released Version 4.10.1 of the plugin with a potential solution. I released it publicly and immediately because, at worst, it is better than Version 4.10 in its handling of "Sticky" themes for those who have caching plugins installed.
In case you wonder about a mysterious keyword=value that appears on URLs after the first sticky query, I had to "hide" the value to avoid confusion occasionally caused by browser caching. The keyword=value was added to make the URL unique so that a Caching plugin will cache each theme's version of the "page" separately.
In the end, I did not use Javascript, but will if this solution does not work out.
Please test thoroughly and let me know the results.
I have some more work to do on this plugin for other users, so you may see new releases come out soon that do not have anything to do with your reported problems with the plugin.