### 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 *scene = editor_data.get_edited_scene_root(scene_idx);
			if (scene && !scene->get_scene_file_path().is_empty()) {
				if (DirAccess::exists(scene->get_scene_file_path().get_base_dir())) {
					if (scene_idx != editor_data.get_edited_scene()) {
						_save_scene_with_preview(scene->get_scene_file_path(), scene_idx);
					} else {
						_save_scene_with_preview(scene->get_scene_file_path());
					}
					if (scene_idx != -1) {
						_discard_changes();
					}
					save_editor_layout_delayed();
				} else {
					show_save_accept(vformat(TTR("%s no longer exists! Please specify a new save location."), scene->get_scene_file_path().get_base_dir()), TTR("OK"));
				}
				break;
			}
			[[fallthrough]];
		}
		case FILE_SAVE_AS_SCENE: {
			int scene_idx = (p_option == FILE_SAVE_SCENE || p_option == FILE_SAVE_AS_SCENE) ? -1 : tab_closing_idx;
			Node *scene = editor_data.get_edited_scene_root(scene_idx);
			if (!scene) {
				if (p_option == FILE_SAVE_SCENE) {
					
					
					ScriptEditor::get_singleton()->save_current_script();
				}
				const int saved = _save_external_resources();
				if (saved > 0) {