### 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
settings_menu->add_shortcut(ED_SHORTCUT("editor/command_palette", TTR("Command Palette..."), KeyModifierMask::CMD_OR_CTRL | KeyModifierMask::SHIFT | Key::P), HELP_COMMAND_PALETTE);
	settings_menu->add_separator();
	editor_layouts = memnew(PopupMenu);
	editor_layouts->set_name("Layouts");
	editor_layouts->set_auto_translate(false);
	settings_menu->add_child(editor_layouts);
	editor_layouts->connect("id_pressed", callable_mp(this, &EditorNode::_layout_menu_option));
	settings_menu->add_submenu_item(TTR("Editor Layout"), "Layouts");
	settings_menu->add_separator();
	ED_SHORTCUT_AND_COMMAND("editor/take_screenshot", TTR("Take Screenshot"), KeyModifierMask::CTRL | Key::F12);
	ED_SHORTCUT_OVERRIDE("editor/take_screenshot", "macos", KeyModifierMask::META | Key::F12);
	settings_menu->add_shortcut(ED_GET_SHORTCUT("editor/take_screenshot"), EDITOR_SCREENSHOT);
	settings_menu->set_item_tooltip(-1, TTR("Screenshots are stored in the Editor Data/Settings Folder."));
#ifndef ANDROID_ENABLED
	ED_SHORTCUT_AND_COMMAND("editor/fullscreen_mode", TTR("Toggle Fullscreen"), KeyModifierMask::SHIFT | Key::F11);
	ED_SHORTCUT_OVERRIDE("editor/fullscreen_mode", "macos", KeyModifierMask::META | KeyModifierMask::CTRL | Key::F);
	settings_menu->add_shortcut(ED_GET_SHORTCUT("editor/fullscreen_mode"), SETTINGS_TOGGLE_FULLSCREEN);
#endif
	settings_menu->add_separator();
#ifndef ANDROID_ENABLED
	if (OS::get_singleton()->get_data_path() == OS::get_singleton()->get_config_path()) {
		
		settings_menu->add_item(TTR("Open Editor Data/Settings Folder"), SETTINGS_EDITOR_DATA_FOLDER);
	} else {
		
		settings_menu->add_item(TTR("Open Editor Data Folder"), SETTINGS_EDITOR_DATA_FOLDER);
		settings_menu->add_item(TTR("Open Editor Settings Folder"), SETTINGS_EDITOR_CONFIG_FOLDER);
	}
	settings_menu->add_separator();
#endif