Does this show that reducing immigration increases wages and lowers unemployment and that immigration and labor markets follow the laws of supply and demand?

It was shown by economists at the Kansas City Fed that during Covid, when immigration restrictions were enacted, real wages increased and unemployment decreased.

Subsequently, when immigration restrictions were lifted, it was shown that increased immigration lowers wage growth and lowers job vacancies.

Does this prove (at least in the short/medium term) that immigration and labor markets follow the laws of supply and demand?