Simultaneous EF1 under two heterogeneous monotone valuation profiles

Investigate whether, for every instance with two heterogeneous monotone valuation profiles over indivisible goods—namely, monotone subjective utilities (u_i) and heterogeneous monotone market valuations (v_i)—there exists an allocation that is envy-free up to one good (EF1) simultaneously with respect to both profiles; further, ascertain whether the existence is guaranteed when one of the two profiles (either (u_i) or (v_i)) consists of additive valuations.

Background

Beyond additivity, the paper explores whether EF1 can be achieved simultaneously when both evaluation profiles are only assumed monotone. While EF1 w.r.t. a single monotone profile is known to exist, introducing a second (market-value) profile creates a significant challenge.

This open question explicitly asks for existence under two heterogeneous monotone profiles and also poses a sub-question for the special case where one profile is additive.