Recall what the spec says about minimized surfaces:

A display surface that is being shared may temporarily or permanently become inaccessible to the application because of actions taken by the operating system or user agent. What makes a display surface considered inaccesible is outside the scope of this specification, but examples MAY include a monitor disconnecting, window or browser closing or becoming minimized, or due to an incoming call on a phone.

So minimizing a window might pause the capture.

Suppose a Web developer wants to warn users if minimizing windows would stop the capture; we have an example of such a developer here:

We need to give the user a hint that minimizing the window will cause the screen to not capture data

Currently, there's no way for this developer to know if the warning should be issued. It would be nice if developers could query that. Ideally before getDisplayMedia() is ever called, but possibly also after, via a MediaStreamTrackCapability.