No, there isn't, and there are very good reasons for doing that:
That's not the "GSX menu", it's the "Simconnect menu", which is shared by ALL 3rd party applications that use it so, if GSX tried to always keep it open, it would prevent every other application from using it, since only one menu at time can be visible and we have no control over this.
The way the Simconnect menu works, requires to close the menu before for an application can be notified WHICH choice you made from it, and it's the simulator itself that closes the menu automatically when you choose an entry. To have the menu always open, would require immediately open it again after it closed. Which again, would prevent any other products from using it.
