### 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
switch (current_menu_option) {
		case FILE_NEW_INHERITED_SCENE: {
			Node *scene = editor_data.get_edited_scene_root();
			
			if (!scene) {
				_menu_option_confirm(FILE_CLOSE, true);
			}
			load_scene(p_file, false, true);
		} break;
		case FILE_OPEN_SCENE: {
			load_scene(p_file);
		} break;
		case SETTINGS_PICK_MAIN_SCENE: {
			ProjectSettings::get_singleton()->set("application/run/main_scene", p_file);
			ProjectSettings::get_singleton()->save();
			
			project_run_bar->play_main_scene((bool)pick_main_scene->get_meta("from_native", false));
		} break;
		case FILE_CLOSE:
		case SCENE_TAB_CLOSE:
		case FILE_SAVE_SCENE:
		case FILE_SAVE_AS_SCENE: {
			int scene_idx = (current_menu_option == FILE_SAVE_SCENE || current_menu_option == FILE_SAVE_AS_SCENE) ? -1 : tab_closing_idx;
			if (file->get_file_mode() == EditorFileDialog::FILE_MODE_SAVE_FILE) {
				bool same_open_scene = false;
				for (int i = 0; i < editor_data.get_edited_scene_count(); i++) {
					if (editor_data.get_scene_path(i) == p_file && i != scene_idx) {
						same_open_scene = true;