### 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
main_plugin->edit(current_obj);
			} else {
				editor_plugin_screen->edit(current_obj);
			}
			is_main_screen_editing = true;
		} else if (!main_plugin && editor_plugin_screen && is_main_screen_editing) {
			editor_plugin_screen->edit(nullptr);
			is_main_screen_editing = false;
		}
		edit_item(current_obj, editor_owner);
	}
	InspectorDock::get_singleton()->update(current_obj);
}
void EditorNode::_android_build_source_selected(const String &p_file) {
	export_template_manager->install_android_template_from_file(p_file);
}
void EditorNode::_menu_option_confirm(int p_option, bool p_confirmed) {
	if (!p_confirmed) { 
		current_menu_option = (MenuOptions)p_option;
	}
	switch (p_option) {
		case FILE_NEW_SCENE: {
			new_scene();
		} break;
		case FILE_NEW_INHERITED_SCENE:
		case FILE_OPEN_SCENE: {
			file->set_file_mode(EditorFileDialog::FILE_MODE_OPEN_FILE);
			List<String> extensions;
			ResourceLoader::get_recognized_extensions_for_type("PackedScene", &extensions);
			file->clear_filters();