Project Manager shows outdated path when renaming lowercase to uppercase

### Tested versions

Reproducible in 4.7.beta [ef02314], 4.6.stable.

### System information

Godot v4.6.stable - Windows 11 (build 26200) - Multi-window, 1 monitor - Direct3D 12 (Forward+) - dedicated NVIDIA GeForce RTX 3060 Laptop GPU (NVIDIA; 32.0.15.9636) - 12th Gen Intel(R) Core(TM) i7-12700H (20 threads) - 15.69 GiB memory

### Issue description

When renaming the project path from lowercase to uppercase, or vice versa, the Project Manager still shows the original path (from when the project was created).

It can still open the project regardless, as the Windows file system is case-insensitive.

### Steps to reproduce

1. Create a new project and quit.
2. Rename the project's folder name to use lowercase/uppercase.
3. Open Godot, and the Project Manager still shows the original path.

Screenshot:

(The actual path is "C:\Users\Edward Chan\Documents\Git\Godot_Test_4_7_Dev" after the rename.)

### Minimal reproduction project (MRP)

N/A