### 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
EditorExport::get_singleton()->add_export_plugin(gdextension_export_plugin);
	Ref<DedicatedServerExportPlugin> dedicated_server_export_plugin;
	dedicated_server_export_plugin.instantiate();
	EditorExport::get_singleton()->add_export_plugin(dedicated_server_export_plugin);
	Ref<PackedSceneEditorTranslationParserPlugin> packed_scene_translation_parser_plugin;
	packed_scene_translation_parser_plugin.instantiate();
	EditorTranslationParser::get_singleton()->add_parser(packed_scene_translation_parser_plugin, EditorTranslationParser::STANDARD);
	_edit_current();
	current = nullptr;
	saving_resource = Ref<Resource>();
	set_process(true);
	open_imported = memnew(ConfirmationDialog);
	open_imported->set_ok_button_text(TTR("Open Anyway"));
	new_inherited_button = open_imported->add_button(TTR("New Inherited"), !DisplayServer::get_singleton()->get_swap_cancel_ok(), "inherit");
	open_imported->connect("confirmed", callable_mp(this, &EditorNode::_open_imported));
	open_imported->connect("custom_action", callable_mp(this, &EditorNode::_inherit_imported));
	gui_base->add_child(open_imported);
	quick_open = memnew(EditorQuickOpen);
	gui_base->add_child(quick_open);
	quick_open->connect("quick_open", callable_mp(this, &EditorNode::_quick_opened));
	_update_recent_scenes();
	set_process_shortcut_input(true);
	load_errors = memnew(RichTextLabel);
	load_error_dialog = memnew(AcceptDialog);
	load_error_dialog->set_unparent_when_invisible(true);
	load_error_dialog->add_child(load_errors);
	load_error_dialog->set_title(TTR("Load Errors"));
	execute_outputs = memnew(RichTextLabel);
	execute_outputs->set_selection_enabled(true);
	execute_outputs->set_context_menu_enabled(true);