Minimal number of goods in a three-agent PMMS counterexample

Determine whether nine goods are the minimum number required for a three-agent additive-valuation instance with no PMMS allocation, or whether a counterexample exists with at most eight goods.

Background

The paper constructs a nonexistence instance for PMMS with three agents and nine goods. It also cites a positive result establishing PMMS existence when the number of goods is at most the number of agents plus two, which rules out three-agent counterexamples with five or fewer goods.

The authors report that they have not found a three-agent counterexample with six, seven, or eight goods, but do not establish that nine goods is minimal. Their stated belief that nine is minimal or almost minimal leaves the exact threshold unresolved.