Option "Always load last project on startup" is broken on UE 5.3

In UE 5.3, when the option "Always load last project on startup" is enabled and a new Editor launch attempts to automatically reopen the last closed project, the dialog "The following modules are missing or built with a different engine version" pops up every time, even if there were no changes to the project. The project ends up not opening with either choice on that dialog, so it must then be opened manually.

Notes:
1) The option can be found in at least 3 places:

• When launching the Editor without specifying a project to open
• Inside the Editor, on "File – Open Project"
• Inside the Editor, on "Edit – Editor Preferences – Loading & Saving – Startup – Load the Most Recently Loaded Project at Startup"

2) The option is relevant when launching the Editor without specifying a project to open, for example through the Epic Games Launcher or by running the executable directly.

3) As reported by the UDN member on the linked case, the problem seems to be rooted in file [Engine\Source\Runtime\Launch\Private\LaunchEngineLoop.cpp]

• [:2103] This condition only passes when a project was specified to be opened on launch.
• [:2105] ProjectBinariesRootDirectory would be correctly set to the absolute path of the project, but this line is not reached.
• [:2586] This condition now passes with the name of the Last Opened Project
• [:2589] This uses ProjectBinariesRootDirectory, which is still empty in this case. As a result, ProjectBinariesDirectory is set to path relative to the uproject file, while the rest of the code expects an absolute path.
• [:6469] This will fail to find the C++ modules and add all of them as incompatible.

4) Up to UE 5.2, lines 2103, 2105 and 2586 above behaved similarly. However, line 2589 used FPlatformMisc::ProjectDir() again, just like line 2105, instead of attempting to use the value cached in ProjectBinariesRootDirectory.

• Open any C++ project inside UE 5.3
• Go to "Edit – Editor Preferences – Loading & Saving" and check "Startup – Load the Most Recently Loaded Project at Startup"
• Close the Editor
• Reopen the Editor through the Epic Games Launcher
• The Editor should now open the same project as before
• Instead, the "missing modules" dialog pops up and the project fails to open
• Next time the Editor is launched, the project will have to be opened manually

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