Determine fixed-parameter tractability in the number of agents

Determine whether Nash welfare maximization for ordered or doubling lexicographic valuations admits an exact algorithm with running time $f(n)\cdot\operatorname{poly}(m)$, or establish that such a fixed-parameter tractable algorithm is unlikely.

Background

The paper gives exact polynomial-time algorithms for ordered and doubling lexicographic valuations when the number of agents is constant. Their running times are XP with respect to the number of agents, meaning that the exponent of the polynomial in the number of goods depends on the number of agents.

The open question asks whether these exact algorithms can be improved to fixed-parameter tractable algorithms whose dependence on the number of agents is confined to a multiplicative function $f(n)$, or whether parameterized-complexity barriers should be expected.