### Tested versions

Godot Engine v4.2.stable.official.46dc27791

### System information

Windows 10

### Issue description

When exporting to web, Godot fails with this error in the console:

> scene/main/window.cpp:884 - Attempting to make child window exclusive, but the parent window already has another exclusive child. This window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559/@<!-- -->VBoxContainer@<!-- -->137/@<!-- -->AcceptDialog@<!-- -->411, parent window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559, current exclusive child window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559/@<!-- -->ProgressDialog@<!-- -->17444

![a](https://github.com/godotengine/godot/assets/2000051/bf9d22aa-d3fe-4e11-bff6-c387e740fe90)

This happens for newly created empty projects.


### Steps to reproduce

1. Open the attached project (or create a new project).
2. Project->Export->Export All
3. Observe the console error.  No output files are generated.


### Minimal reproduction project (MRP)

[export_test.zip](https://github.com/godotengine/godot/files/13574810/export_test.zip)

String project_icon_path = GLOBAL_GET("application/config/icon");
	icon.instantiate();
	foreground.instantiate();
	background.instantiate();
	
	String path = static_cast<String>(p_preset->get(launcher_icon_option)).strip_edges();
	print_verbose("Loading regular icon from " + path);
	if (path.is_empty() || ImageLoader::load_image(path, icon) != OK) {
		print_verbose("- falling back to project icon: " + project_icon_path);
		ImageLoader::load_image(project_icon_path, icon);
	}
	
	path = static_cast<String>(p_preset->get(launcher_adaptive_icon_foreground_option)).strip_edges();
	print_verbose("Loading adaptive foreground icon from " + path);
	if (path.is_empty() || ImageLoader::load_image(path, foreground) != OK) {
		print_verbose("- falling back to using the regular icon");
		foreground = icon;
	}
	
	path = static_cast<String>(p_preset->get(launcher_adaptive_icon_background_option)).strip_edges();
	if (!path.is_empty()) {
		print_verbose("Loading adaptive background icon from " + path);
		ImageLoader::load_image(path, background);
	}
}
void EditorExportPlatformAndroid::store_image(const LauncherIcon launcher_icon, const Vector<uint8_t> &data) {