I can reproduce a very similar intermittent connection pattern on Opera One Stable 136.0.6008.52.
Environment:
Windows 10 Pro 22H2, OS Build 19045.6456
Opera One Stable 136.0.6008.52
Browser Connector enabled and already connected to ChatGPT
In a fresh test after the connector was already working, I observed this sequence:
list-tabs -> success
tab-content -> success
list-tabs -> "Browser not connected"
history -> "Browser not connected"
structured tab-content query -> success
list-tabs -> success
tab-content -> success
list-tabs -> "Browser not connected"
I did not reconnect the Browser Connector, change permissions, restart Opera, or sign in again between these calls.
So in my case the connection is not permanently lost: it can fail, recover automatically without user action, and then fail again shortly afterward.
I also reproduced the same behavior before and after updating Opera to 136.0.6008.52.
I have submitted the same finding through Opera's official Desktop Stable bug-report form as well.