Hmm, I’m not seeing the issue you are describing. I have seen something similar is Firefox when using form elements – a refresh would not reset the input values and cause a whole bunch of problems – but nothing like this with class names.
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This is actually standard browser behavior and not an issue with jQuery DOM cache. For anyone interested, I solved this by adding the following code to jQuery. Essentially it will reset the page/scroll position to the top after a refresh.