### 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
DEV_ASSERT(!singleton);
	singleton = this;
	Resource::_get_local_scene_func = _resource_get_edited_scene;
	{
		PortableCompressedTexture2D::set_keep_all_compressed_buffers(true);
		RenderingServer::get_singleton()->set_debug_generate_wireframes(true);
		AudioServer::get_singleton()->set_enable_tagging_used_audio_streams(true);
		
		if (NavigationServer3D::get_singleton()->get_debug_enabled()) {
			NavigationServer3D::get_singleton()->set_active(true);
		} else {
			NavigationServer3D::get_singleton()->set_active(false);
		}
		
		PhysicsServer3D::get_singleton()->set_active(false);
		PhysicsServer2D::get_singleton()->set_active(false);
		
		ScriptServer::set_scripting_enabled(false);
		Input::get_singleton()->set_use_accumulated_input(true);
		if (!DisplayServer::get_singleton()->is_touchscreen_available()) {
			
			Input::get_singleton()->set_emulate_touch_from_mouse(false);
		}
		DisplayServer::get_singleton()->cursor_set_custom_image(Ref<Resource>());