### 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
Node *p_edited_scene,
		Node *p_root,
		Node *p_node,
		HashMap<NodePath, ModificationNodeEntry> &p_modification_table,
		List<AdditiveNodeEntry> &p_addition_list) {
	bool node_part_of_subscene = p_node != p_edited_scene &&
			p_edited_scene->get_scene_inherited_state().is_valid() &&
			p_edited_scene->get_scene_inherited_state()->find_node_by_path(p_edited_scene->get_path_to(p_node)) >= 0;
	
	Node *valid_node_owner = p_node->get_owner();
	while (valid_node_owner) {
		if (valid_node_owner == p_root) {
			break;
		}
		valid_node_owner = valid_node_owner->get_owner();
	}
	if ((valid_node_owner == p_root && (p_root != p_edited_scene || !p_edited_scene->get_scene_file_path().is_empty())) || node_part_of_subscene || p_node == p_root) {
		HashMap<StringName, Variant> modified_properties = get_modified_properties_for_node(p_node);
		
		List<Connection> connections_to;
		p_node->get_all_signal_connections(&connections_to);
		List<ConnectionWithNodePath> valid_connections_to;
		for (const Connection &c : connections_to) {
			Node *connection_target_node = Object::cast_to<Node>(c.callable.get_object());