So the monitor shows that message, or the browser on the monitor shows that message. Monitors don't know about a network, FYI!
It loks like your network shuts down after going into power saving mode. Why don't you set the PC to not go into power saving mode. Do you know where those settings are?
This isn't a monitor issue, this is a PC configuration issue, and it is working as expected based on how your PC is set up for sleeping and power saving. What Operating system is it?