### 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
EditorInspector::cleanup_plugins();
	EditorTranslationParser::get_singleton()->clean_parsers();
	ResourceImporterScene::clean_up_importer_plugins();
	remove_print_handler(&print_handler);
	EditorHelp::cleanup_doc();
	memdelete(editor_selection);
	memdelete(editor_plugins_over);
	memdelete(editor_plugins_force_over);
	memdelete(editor_plugins_force_input_forwarding);
	memdelete(progress_hb);
	memdelete(surface_upgrade_tool);
	EditorSettings::destroy();
	EditorColorMap::finish();
	EditorTheme::finalize();
	GDExtensionEditorPlugins::editor_node_add_plugin = nullptr;
	GDExtensionEditorPlugins::editor_node_remove_plugin = nullptr;
	FileDialog::get_icon_func = nullptr;
	FileDialog::register_func = nullptr;
	FileDialog::unregister_func = nullptr;
	EditorFileDialog::get_icon_func = nullptr;
	EditorFileDialog::register_func = nullptr;
	EditorFileDialog::unregister_func = nullptr;
	file_dialogs.clear();
	editor_file_dialogs.clear();
	singleton = nullptr;
}
/*
 * EDITOR PLUGIN LIST
 */
void EditorPluginList::make_visible(bool p_visible) {