Embedded game does not use project icon on Linux/X11 (fallback X11 icon is used instead)

### Tested versions

- Reproducible in: 4.4.1.stable, 4.5.dev4

### System information

Godot v4.5.dev (bc6f0d29c) - Fedora Linux 42 (KDE Plasma Desktop Edition) on X11 - X11 display driver, Multi-window, 1 monitor - Vulkan (Forward+) - dedicated NVIDIA GeForce RTX 5090 (nvidia; 570.144) - AMD Ryzen 9 9950X3D 16-Core Processor (32 threads)

### Issue description

Embedded game does not use project icon on Linux/X11 (fallback X11 icon is used instead):

### Game embedding enabled


### Game embedding disabled


### Steps to reproduce

- Configure `application/config/icon` in the Project Settings.
- Run the project and alt-tab. Look at the icon in the window border and alt-tab window switcher.
- Stop the project, disable **Embed Game on Next Play** in the Game tab, then run the project again.
- Look at the window border icon and alt-tab icon.

### Minimal reproduction project (MRP)

Any project will do; I've used https://github.com/godotengine/godot-demo-projects/tree/master/3d/tonemap_color_correction for testing.