I want to book an Open Jaw ticket in United Airlines' website that looks like this:

Frankfurt -> USA (Operated by United Airlines)
Mexico -> Frankfurt (Operated by Lufthansa)

I can choose to book the ticket in the "Basic Economy" class which does not offer a checked luggage normally.
United's website says that for *A Gold passenger, for flights operated by United, a free checked luggage is offered.

My question now is what happen will happen to Mexico->Frankfurt part, since it is not operated by United. As far as I know
Swiss and Lufthansa do not offer a free baggage for "Economy Light" fares. Will Lufthansa treat United's "Basic Economy"
as equivalen to Lufthansa's "Economy Light" and therefore I will not qualify for a free luggage?