To reproduce:
1. Open project attached by QA
2. Make sure that your device is disconnected from the Game Center (Settings -> Game Center)
3. Build and run "test" scene on iOS
4. Press the button
5. Notice that game center window has appeared
6. Click "cancel"
7. Hit the button once again

Expected result: Game Center window should appear again.
Actual result: Game Center window is not being shown.

Reproduced with: Unity 2017.1.3p2, 2017.2.0p4, 2017.3.1p2, 2018.2.0a5
Devices: iPhone 8 iOS 11.0, iPod Touch 6th gen iOS 10.2, iPad Mini 2 iOS 9.2.1

Won't fix:
GameCenter API works in such that that the authentication window is no longer shown if the user cancels it. The displayed error ("Failed to authenticate local user The requested operation has been canceled or disabled by the user.") is returned by `GKLocalPlayer.authenticateHandler' so there is nothing we can really do if the iOS doesn't show the dialog.