### 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
for (int i = 0; i < editor_data.get_edited_scene_count(); i++) {
		Node *scene = editor_data.get_edited_scene_root(i);
		if (scene && p_scene_paths.has(scene->get_scene_file_path())) {
			_save_scene(scene->get_scene_file_path(), i);
		}
	}
}
void EditorNode::save_before_run() {
	current_menu_option = FILE_SAVE_AND_RUN;
	_menu_option_confirm(FILE_SAVE_AS_SCENE, true);
	file->set_title(TTR("Save scene before running..."));
}
void EditorNode::try_autosave() {
	if (!bool(EDITOR_GET("run/auto_save/save_before_running"))) {
		return;
	}
	if (unsaved_cache) {
		Node *scene = editor_data.get_edited_scene_root();
		if (scene && !scene->get_scene_file_path().is_empty()) { 
			_save_scene_with_preview(scene->get_scene_file_path());
		}
	}
	_menu_option(FILE_SAVE_ALL_SCENES);
	editor_data.save_editor_external_data();
}
void EditorNode::restart_editor() {