### 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
distraction_free = memnew(Button);
	distraction_free->set_flat(true);
	ED_SHORTCUT_AND_COMMAND("editor/distraction_free_mode", TTR("Distraction Free Mode"), KeyModifierMask::CTRL | KeyModifierMask::SHIFT | Key::F11);
	ED_SHORTCUT_OVERRIDE("editor/distraction_free_mode", "macos", KeyModifierMask::META | KeyModifierMask::CTRL | Key::D);
	distraction_free->set_shortcut(ED_GET_SHORTCUT("editor/distraction_free_mode"));
	distraction_free->set_tooltip_text(TTR("Toggle distraction-free mode."));
	distraction_free->set_toggle_mode(true);
	scene_tabs->add_extra_button(distraction_free);
	distraction_free->connect("pressed", callable_mp(this, &EditorNode::_toggle_distraction_free_mode));
	scene_root_parent = memnew(PanelContainer);
	scene_root_parent->set_custom_minimum_size(Size2(0, 80) * EDSCALE);
	scene_root_parent->add_theme_style_override("panel", theme->get_stylebox(SNAME("Content"), EditorStringName(EditorStyles)));
	scene_root_parent->set_draw_behind_parent(true);
	srt->add_child(scene_root_parent);
	scene_root_parent->set_v_size_flags(Control::SIZE_EXPAND_FILL);
	scene_root = memnew(SubViewport);
	scene_root->set_embedding_subwindows(true);
	scene_root->set_disable_3d(true);
	scene_root->set_disable_input(true);
	scene_root->set_as_audio_listener_2d(true);
	main_screen_vbox = memnew(VBoxContainer);
	main_screen_vbox->set_name("MainScreen");
	main_screen_vbox->set_v_size_flags(Control::SIZE_EXPAND_FILL);
	main_screen_vbox->add_theme_constant_override("separation", 0);
	scene_root_parent->add_child(main_screen_vbox);
	bool global_menu = !bool(EDITOR_GET("interface/editor/use_embedded_menu")) && DisplayServer::get_singleton()->has_feature(DisplayServer::FEATURE_GLOBAL_MENU);
	bool can_expand = bool(EDITOR_GET("interface/editor/expand_to_title")) && DisplayServer::get_singleton()->has_feature(DisplayServer::FEATURE_EXTEND_TO_TITLE);
	if (can_expand) {
		
		left_menu_spacer = memnew(Control);