### 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
List<Node *> nodes_owned_by_original_node;
				original_node->get_owned_by(original_node, &nodes_owned_by_original_node);
				for (Node *owned_node : nodes_owned_by_original_node) {
					owned_node->set_owner(nullptr);
				}
				
				while (original_node->get_child_count()) {
					Node *child = original_node->get_child(0);
					original_node->remove_child(child);
					child->queue_free();
				}
				
				bool is_editable = true;
				Node *owner = original_node->get_owner();
				if (owner) {
					is_editable = owner->is_editable_instance(original_node);
				}
				
				Ref<PackedScene> current_packed_scene;
				if (original_node->get_scene_file_path() == p_instance_path) {
					
					
					current_packed_scene = instance_scene_packed_scene;
				} else {
					
					
					List<String> required_load_paths;
					String scene_path = original_node->get_scene_file_path();