Does k-SUM/XOR imply the Primal Pathwidth SETH?

Establish whether the k-SUM Hypothesis (for all fixed k ≥ 3) or the k-XOR Hypothesis implies the Primal Pathwidth SETH, i.e., that for every epsilon > 0 there is no algorithm that, given a CNF formula together with a path decomposition of the primal graph of width pw, decides satisfiability in time (2−epsilon)^pw · |phi|^{O(1)}.

Background

The paper proves that k-SUM (and k-XOR) hardness implies the Primal Treewidth SETH, by giving randomized reductions from k-SUM/k-XOR to SAT instances of treewidth about (k/2)·log n.

The authors note their construction crucially exploits treewidth (allowing alternations via join nodes) rather than pathwidth (which corresponds to nondeterministic log-space without alternations). They explicitly leave open whether similar reductions can be made to work with pathwidth, which would yield an implication to the Primal Pathwidth SETH.

A positive resolution would strengthen their main implication, while a negative resolution would provide evidence that the Primal Treewidth SETH and the Primal Pathwidth SETH are not equivalent.