### 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
layout_dialog = memnew(EditorLayoutsDialog);
	gui_base->add_child(layout_dialog);
	layout_dialog->set_hide_on_ok(false);
	layout_dialog->set_size(Size2(225, 270) * EDSCALE);
	layout_dialog->connect("name_confirmed", callable_mp(this, &EditorNode::_dialog_action));
	update_spinner = memnew(MenuButton);
	right_menu_hb->add_child(update_spinner);
	update_spinner->set_icon(theme->get_icon(SNAME("Progress1"), EditorStringName(EditorIcons)));
	update_spinner->get_popup()->connect("id_pressed", callable_mp(this, &EditorNode::_menu_option));
	PopupMenu *p = update_spinner->get_popup();
	p->add_radio_check_item(TTR("Update Continuously"), SETTINGS_UPDATE_CONTINUOUSLY);
	p->add_radio_check_item(TTR("Update When Changed"), SETTINGS_UPDATE_WHEN_CHANGED);
	p->add_separator();
	p->add_item(TTR("Hide Update Spinner"), SETTINGS_UPDATE_SPINNER_HIDE);
	_update_update_spinner();
	
	memnew(SceneTreeDock(scene_root, editor_selection, editor_data));
	memnew(InspectorDock(editor_data));
	memnew(ImportDock);
	memnew(NodeDock);
	FileSystemDock *filesystem_dock = memnew(FileSystemDock);
	filesystem_dock->connect("inherit", callable_mp(this, &EditorNode::_inherit_request));
	filesystem_dock->connect("instantiate", callable_mp(this, &EditorNode::_instantiate_request));
	filesystem_dock->connect("display_mode_changed", callable_mp(this, &EditorNode::_save_editor_layout));
	get_project_settings()->connect_filesystem_dock_signals(filesystem_dock);
	history_dock = memnew(HistoryDock);
	
	dock_slot[DOCK_SLOT_LEFT_UR]->add_child(SceneTreeDock::get_singleton());
	dock_slot[DOCK_SLOT_LEFT_UR]->set_tab_title(dock_slot[DOCK_SLOT_LEFT_UR]->get_tab_idx_from_control(SceneTreeDock::get_singleton()), TTR("Scene"));