All I can think of is: there is a meta HTML tag, or HTTP header, that can be included in or with a web page that will prevent browsers from caching that page, so you always get the latest version of it and shouldn't have to refresh it. But as far as I know, there is nothing like that for individual images. I could be mistaken.
Anyway, you should only have to do a refresh once after replacing your profile image; and most people don't do that very often, so it wouldn't be a major problem.
Installing and upgrading help
Profile images won't display
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