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Hey OP, I have experienced the same issue. My website always showed the home page cached with no changes on purge. This is how I solved it :

Go to litespeed cache > page optimisation > CSS Settings

1. css combine > off
2. CSS Combine External and Inline > off
3. Load CSS Asynchronously > off
4. CCSS per url > off

Then in JS settings

1. JS Combine > off
2. JS Combine External and Inline > off

Keep everything on in both css and js settings except the above mentioned one.

Go to litespeed cache > toolbox

1. Purge Homepage
2. Purge All
3. Test the site after 2 or 3 minutes in incognito mode to see if the homepage updates or not.

The above steps worked for me.