Combined parameterization by partial-order width and graph width

Determine the complexity of the Partially Ordered Hamiltonian Path Problem when parameterized jointly by the width of the partial order and a graph width parameter such as treewidth or distance to clique.

Background

Earlier work cited by the paper establishes that POHPP is fixed-parameter tractable and hard, respectively, under parameterization by the width of the partial order. The paper leaves unresolved how combining this order-theoretic parameter with graph structure affects complexity, explicitly mentioning treewidth and distance to clique as candidate graph width parameters.