I had something like that happen a few years ago. If I became inactive for several seconds, it would disconnect. The problem lasted for several months and then magically stopped on its own. I never discovered the cause or a real solution. As a workaround, I wrote a script that would Look every 13 seconds. I ran it when I planned to be inactive. I don't remember why I chose 13 seconds.

Start:
put look
pause 13
goto Start

If that works, then, in my completely unqualified opinion, it is a software issue. If the script does not keep you connected, then there may be a problem with your internet connection, causing you to briefly disconnect at random times. You might not notice those brief disconnections when using other websites or even when streaming online media. If you haven't already, you could try to log into your internet router and review the logs to see if you are losing your connection to the internet. The theory about internet disconnection does not explain why it only happens when you are inactive, so I fear it is the harder to diagnose problem of a software conflict.