S2721DGF, two computers, switching off inactive input?

I have a S2721DGF monitor. I have a work laptop and a home laptop. Both are running Windows.

For my work setup, I have the laptop closed, so I am not using the laptop screen. I have two monitors connected to it. One is the S2721DGF via HDMI and the other is some old monitor.

For my home setup, I use the laptop as a screen and the S2721DGF via DisplayPort as the other monitor. So both computers share the S2721DGF.

My question is this. When I have the input for the S2721DGF set to my work monitor, my home laptop still thinks two monitors are connected to it. I can move my mouse to the monitor and the applications are still on that screen even though I can't see it. This is also true for the work laptop when the input is set to the home laptop. That sounds way more confusing than it is. Anyway, this is my question. Is there a way to make the the computer that isn't the selected input act as though it isn't even physically connected to the laptop? When it isn't the selected one, I want it to act like I actually unplugged the cable. It seems strange that the computers act like they have a second monitor attached if the input is set to another computer.

Thanks.