### 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
_update_theme(true);
			OS::get_singleton()->set_low_processor_usage_mode_sleep_usec(int(EDITOR_GET("interface/editor/low_processor_mode_sleep_usec")));
			get_tree()->get_root()->set_as_audio_listener_3d(false);
			get_tree()->get_root()->set_as_audio_listener_2d(false);
			get_tree()->get_root()->set_snap_2d_transforms_to_pixel(false);
			get_tree()->get_root()->set_snap_2d_vertices_to_pixel(false);
			get_tree()->set_auto_accept_quit(false);
#ifdef ANDROID_ENABLED
			get_tree()->set_quit_on_go_back(false);
#endif
			get_tree()->get_root()->connect("files_dropped", callable_mp(this, &EditorNode::_dropped_files));
			command_palette->register_shortcuts_as_command();
			MessageQueue::get_singleton()->push_callable(callable_mp(this, &EditorNode::_begin_first_scan));
			
		} break;
		case NOTIFICATION_EXIT_TREE: {
			if (progress_dialog) {
				progress_dialog->queue_free();
			}
			if (load_error_dialog) {
				load_error_dialog->queue_free();
			}
			if (execute_output_dialog) {
				execute_output_dialog->queue_free();
			}
			if (warning) {
				warning->queue_free();
			}
			if (accept) {
				accept->queue_free();