Assuming you're using WebRTC - and if so, unfortunately in my experience there's no way to continue the call.
Even if persistent connection is enabled, a refresh of the Genesys desktop application or relevant browser tab will terminate the WebRTC session. Only way around this is to use Remote Number or a physical phone (that doesn't rely on the app/browser handling the call traffic).
I wish Genesys would hang onto the call at the system level - so if a crash or refresh of the browser drops the WebRTC session, there's the opportunity to resume the call or do something else with it (i.e. return to queue, send to voicemail, play a message before disconnecting, etc)
I've created an idea just for this:
Auto-hold WebRTC calls during refresh/crash/dropout, offer recovery options (resume on reload, return to queue, etc)
https://genesyscloud.ideas.aha.io/ideas/CEV-I-588
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Jeff
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