Can you elaborate on "loses the ArcGIS Server connection"? ArcGIS Pro can connect to ArcGIS Enterprise for a variety of needs/uses, and those connections are on different screens. By reconnecting, do you mean you have to authenticate again? If so, what are you authenticating for? Licensing, consuming GIS services, etc...?
I seem to have solved my own issue cause I opened Pro this morning and my connection is still there.
Yesterday when I was attempting to do this, I was right clicking on Project in the Catalog Window, then I would click on New ArcGIS Server Connection, enter the URL to my server and it would connect but when I closed Pro and then reopen it again the connection wasn't there and I had to go through the process again to reconnect.
How Pro handles connections, not just GIS Server connections is a bit goofy in my mind. When you create a new GIS Server connection in Pro, it is put in the "Servers" folder in the Catalog View. The "Servers" folder isn't an actual folder on the file system, it is virtual folder created in the application. In reality, the GIS Server connection is created on-disk in the root folder of the Project. If you create the connection but don't save the Project, the AGS file is still present in the Project root folder on the file system, but it won't be there in the "Servers" virtual folder in all circumstance. You can navigate to the Project's root folder and add it back into the project.