### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
disk_changed_list->clear();
	TreeItem *r = disk_changed_list->create_item();
	disk_changed_list->set_hide_root(true);
	bool need_reload = false;
	
	for (int i = 0; i < editor_data.get_edited_scene_count(); i++) {
		Ref<DirAccess> da = DirAccess::create(DirAccess::ACCESS_RESOURCES);
		if (editor_data.get_scene_path(i) == "" || !da->file_exists(editor_data.get_scene_path(i))) {
			continue;
		}
		uint64_t last_date = editor_data.get_scene_modified_time(i);
		uint64_t date = FileAccess::get_modified_time(editor_data.get_scene_path(i));
		if (date > last_date) {
			TreeItem *ti = disk_changed_list->create_item(r);
			ti->set_text(0, editor_data.get_scene_path(i).get_file());
			need_reload = true;
		}
	}
	String project_settings_path = ProjectSettings::get_singleton()->get_resource_path().path_join("project.godot");
	if (FileAccess::get_modified_time(project_settings_path) > ProjectSettings::get_singleton()->get_last_saved_time()) {
		TreeItem *ti = disk_changed_list->create_item(r);
		ti->set_text(0, "project.godot");
		need_reload = true;
	}
	if (need_reload) {
		disk_changed->call_deferred(SNAME("popup_centered_ratio"), 0.3);
	}
}
void EditorNode::_resave_scenes(String p_str) {