Before revision:  We consider various fairness definitions from fair allocation literature, such as envy-freeness (EF) and leximin optimal fairness. We find that EF and its weaker versions are incompatible with stability, even under a restricted setting with isometric utilities. We focus on leximin optimal fairness and show that finding such a matching is NP-Hard, even under isometric utilities.
After revision:  In particular, we focus on leximin optimal fairness and show that finding such a matching is NP-Hard, even under isometric utilities.