### 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
case RenderingDevice::DeviceType::DEVICE_TYPE_VIRTUAL_GPU:
			device_type_string = "virtual";
			break;
		case RenderingDevice::DeviceType::DEVICE_TYPE_CPU:
			device_type_string = "(software emulation on CPU)";
			break;
		case RenderingDevice::DeviceType::DEVICE_TYPE_OTHER:
		case RenderingDevice::DeviceType::DEVICE_TYPE_MAX:
			break; 
	}
	const Vector<String> video_adapter_driver_info = OS::get_singleton()->get_video_adapter_driver_info();
	const String processor_name = OS::get_singleton()->get_processor_name();
	const int processor_count = OS::get_singleton()->get_processor_count();
	
	if (rendering_method == "forward_plus") {
		rendering_method = "Forward+";
	} else if (rendering_method == "mobile") {
		rendering_method = "Mobile";
	} else if (rendering_method == "gl_compatibility") {
		rendering_method = "Compatibility";
		driver_name = GLOBAL_GET("rendering/gl_compatibility/driver");
	}
	if (driver_name == "vulkan") {
		driver_name = "Vulkan";
	} else if (driver_name.begins_with("opengl3")) {
		driver_name = "GLES3";
	}
	
	Vector<String> info;
	info.push_back(godot_version);