Here is an excerpt from the F5 Local Traffic Manager and DNS Operations Guide.
In order to monitor the health of objects in the configuration, BIG-IP DNS devices in the synchronization group sends monitor requests using iQuery to another iQuery server that is closer to the target of the monitor. All BIG-IP
DNS devices in the synchronization group agrees on which BIG-IP DNS is responsible for initiating the monitoring
request. The result of the monitoring request is sent by the iQuery server to all BIG-IP DNS devices connected to it
that are participating in the synchronization group.
This shows that each gtm is responsible for certain virtual servers and will not send a monitoring request if that virtual server is shown as down. In your case, this means that as long as the BIG-IP DNS devices are in sync, the virtual server status will be consistent among all of the BIG-IP DNS devices so your virtual server would show as down.
If you have any more questions, I am sure I can help.