While I do not advise people using POP (POP3) connection, unless they really
must (as in case some stubborn providers), - I just discovered that Gmail has a feature for multiple POP clients: "recent mode". It overcomes the fact that Gmail doesn't show messages for multiple clients over POP. So, I am posting this just in case someone needs it:
https://support.google.com/mail/answer/47948?hl=en
