### 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
}
				}
				if (same_open_scene) {
					show_warning(TTR("Can't overwrite scene that is still open!"));
					return;
				}
				save_default_environment();
				_save_scene_with_preview(p_file, scene_idx);
				_add_to_recent_scenes(p_file);
				save_editor_layout_delayed();
				if (scene_idx != -1) {
					_discard_changes();
				} else {
					
					editor_data.set_scene_path(editor_data.get_edited_scene(), p_file);
				}
			}
		} break;
		case FILE_SAVE_AND_RUN: {
			if (file->get_file_mode() == EditorFileDialog::FILE_MODE_SAVE_FILE) {
				save_default_environment();
				_save_scene_with_preview(p_file);
				project_run_bar->play_custom_scene(p_file);
			}
		} break;
		case FILE_SAVE_AND_RUN_MAIN_SCENE: {
			ProjectSettings::get_singleton()->set("application/run/main_scene", p_file);
			ProjectSettings::get_singleton()->save();
			if (file->get_file_mode() == EditorFileDialog::FILE_MODE_SAVE_FILE) {
				save_default_environment();