### 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
String scene_file_path = p_node->get_scene_file_path();
	
	if (p_node->get_scene_file_path() == p_instance_path) {
		p_instance_list.push_back(p_node);
	} else {
		Node *current_node = p_node;
		Ref<SceneState> inherited_state = current_node->get_scene_inherited_state();
		while (inherited_state.is_valid()) {
			String inherited_path = inherited_state->get_path();
			if (inherited_path == p_instance_path) {
				p_instance_list.push_back(p_node);
				break;
			}
			inherited_state = inherited_state->get_base_scene_state();
		}
	}
	for (int i = 0; i < p_node->get_child_count(); i++) {
		Node *child = p_node->get_child(i);
		find_all_instances_inheriting_path_in_node(p_root, child, p_instance_path, p_instance_list);
	}
}
void EditorNode::reload_instances_with_path_in_edited_scenes(const String &p_instance_path) {
	int original_edited_scene_idx = editor_data.get_edited_scene();
	HashMap<int, List<Node *>> edited_scene_map;