### 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)

Ref<EditorExportPreset> preset;
	for (int i = 0; i < EditorExport::get_singleton()->get_export_preset_count(); i++) {
		Ref<EditorExportPreset> ep = EditorExport::get_singleton()->get_export_preset(i);
		if (ep->is_runnable() && ep->get_platform() == this) {
			preset = ep;
			break;
		}
	}
	int prev = menu_options;
	menu_options = preset.is_valid();
	if (server->is_listening()) {
		if (menu_options == 0) {
			MutexLock lock(server_lock);
			server->stop();
		} else {
			menu_options += 1;
		}
	}
	return menu_options != prev;
}
Ref<ImageTexture> EditorExportPlatformWeb::get_option_icon(int p_index) const {