Reading through the thread, I think there is enough evidence to point to the problem.
-Your mail client has issues reconnecting to the server after a period of inactivity
-Other clients do not ( Thunderbird )
-Webmail is not comparing apples with apples, but tells us that:
The webmail server does not have problems connecting to the mail server.
Your PC does not have issues connecting to other Internet services, eg HTTP, using a modern browser
This all points to your mail client not handling network disconnections gracefully.
Back In The Day, when a network was a few machines connected with a co-axial cable, things were easier.
A client could open a connection, and assume it was stable forever.
Things move on.
The modern network environment is very different.
Busy servers will disconnect idle clients after a short period.
Battery-powered devices have led to OSes that do power management, and will shut down network interfaces.
Mobile devices will drop on and off between networks as they roam around.
All this will break clients designed on the old paradigm I described.
Modern clients need to work with the OS to deal with this ever-changing network environment.
It seems to me that your email client is failing to gracefully handle one or more of the issues that are simply part of modern network life.