### 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)

EditorExportPreset::FileExportMode file_export_mode = (EditorExportPreset::FileExportMode)(int)p_item->get_metadata(1);
	bool is_inherited = false;
	if (file_export_mode == EditorExportPreset::MODE_FILE_NOT_CUSTOMIZED) {
		file_export_mode = p_inherited_export_mode;
		is_inherited = true;
	}
	if (file_export_mode == EditorExportPreset::MODE_FILE_NOT_CUSTOMIZED) {
		p_item->set_text(1, "");
	} else {
		String text = file_mode_popup->get_item_text(file_mode_popup->get_item_index(file_export_mode));
		if (is_inherited) {
			text += " " + TTR("(Inherited)");
		}
		p_item->set_text(1, text);
	}
	for (int i = 0; i < p_item->get_child_count(); i++) {
		_propagate_file_export_mode(p_item->get_child(i), file_export_mode);
	}
}
void ProjectExportDialog::_tree_changed() {