Balanced EF1 under a single monotone valuation profile

Establish whether, for every instance of fair division of indivisible goods under a single heterogeneous monotone valuation profile (u_i), there exists a balanced allocation—each agent receives either ⌊m/n⌋ or ⌈m/n⌉ goods—that is envy-free up to one good (EF1) with respect to (u_i).

Background

The authors note that EF1 under monotone valuations is known, but achieving balancedness simultaneously with EF1 remains elusive. Balancedness corresponds to EF1 when market values are identical, making this question fundamental even without market values.

This question asks whether one can always guarantee balanced EF1 under general monotone (not necessarily additive) valuations, a core unresolved issue in discrete fair division.